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343 | Simplify the following expression: $x = \dfrac{2}{3a - 2} \times \dfrac{4a}{7}$ | When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ 2 \times 4a } { (3a - 2) \times 7}$ $x = \dfrac{8a}{21a - 14}$ |
343 | Simplify the following expression: $a = \dfrac{10}{x - 2} \times \dfrac{8x}{4}$ | When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ 10 \times 8x } { (x - 2) \times 4}$ $a = \dfrac{80x}{4x - 8}$ Simplify: $a = \dfrac{20x}{x - 2}$ |
343 | Simplify the following expression: $z = \dfrac{r}{10} \div \dfrac{8r}{7}$ | Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{r}{10} \times \dfrac{7}{8r}$ When multiplying fractions, we multiply the numerators and the denominators. $z = \dfrac{ r \times 7 } { 10 \times 8r}$ $z = \dfrac{7r}{80r}$ Simplify: $z = \dfrac{7}{80}$ |
343 | Simplify the following expression: $y = \dfrac{3a - 7}{4} \div \dfrac{3a}{3}$ | Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{3a - 7}{4} \times \dfrac{3}{3a}$ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ (3a - 7) \times 3 } { 4 \times 3a}$ $y = \dfrac{9a - 21}{12a}$ Simplify: $y = \dfrac{3a - 7}{4a}$ |
343 | Simplify the following expression: $q = \dfrac{9}{5t} \times \dfrac{10t}{5}$ | When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ 9 \times 10t } { 5t \times 5}$ $q = \dfrac{90t}{25t}$ Simplify: $q = \dfrac{18}{5}$ |
343 | Simplify the following expression: $q = \dfrac{3t - 3}{8} \times \dfrac{4}{4t}$ | When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ (3t - 3) \times 4 } { 8 \times 4t}$ $q = \dfrac{12t - 12}{32t}$ Simplify: $q = \dfrac{3t - 3}{8t}$ |
343 | Simplify the following expression: $y = \dfrac{8}{6p} \times \dfrac{6p}{8}$ | When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ 8 \times 6p } { 6p \times 8}$ $y = \dfrac{48p}{48p}$ Simplify: $y = \dfrac{1}{1}$ |
343 | Simplify the following expression: $x = \dfrac{3}{8q - 7} \div \dfrac{4}{8q}$ | Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{3}{8q - 7} \times \dfrac{8q}{4}$ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ 3 \times 8q } { (8q - 7) \times 4}$ $x = \dfrac{24q}{32q - 28}$ Simplify: $x = \dfrac{6q}{8q - 7}$ |
343 | Simplify the following expression: $n = \dfrac{3}{2y + 8} \times \dfrac{8y}{4}$ | When multiplying fractions, we multiply the numerators and the denominators. $n = \dfrac{ 3 \times 8y } { (2y + 8) \times 4}$ $n = \dfrac{24y}{8y + 32}$ Simplify: $n = \dfrac{3y}{y + 4}$ |
343 | Simplify the following expression: $p = \dfrac{5n + 10}{8} \times \dfrac{5}{7n}$ | When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ (5n + 10) \times 5 } { 8 \times 7n}$ $p = \dfrac{25n + 50}{56n}$ |
343 | Simplify the following expression: $y = \dfrac{6q - 8}{4} \div \dfrac{10q}{8}$ | Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{6q - 8}{4} \times \dfrac{8}{10q}$ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ (6q - 8) \times 8 } { 4 \times 10q}$ $y = \dfrac{48q - 64}{40q}$ Simplify: $y = \dfrac{6q - 8}{5q}$ |
343 | Simplify the following expression: $t = \dfrac{3n}{2} \div \dfrac{6n}{2}$ | Dividing by an expression is the same as multiplying by its inverse. $t = \dfrac{3n}{2} \times \dfrac{2}{6n}$ When multiplying fractions, we multiply the numerators and the denominators. $t = \dfrac{ 3n \times 2 } { 2 \times 6n}$ $t = \dfrac{6n}{12n}$ Simplify: $t = \dfrac{1}{2}$ |
343 | Simplify the following expression: $y = \dfrac{2}{2p - 9} \div \dfrac{6}{8p}$ | Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{2}{2p - 9} \times \dfrac{8p}{6}$ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ 2 \times 8p } { (2p - 9) \times 6}$ $y = \dfrac{16p}{12p - 54}$ Simplify: $y = \dfrac{8p}{6p - 27}$ |
343 | Simplify the following expression: $a = \dfrac{6x}{9} \div \dfrac{2x}{7}$ | Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{6x}{9} \times \dfrac{7}{2x}$ When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ 6x \times 7 } { 9 \times 2x}$ $a = \dfrac{42x}{18x}$ Simplify: $a = \dfrac{7}{3}$ |
343 | Simplify the following expression: $z = \dfrac{2}{8a} \div \dfrac{10}{8a}$ | Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{2}{8a} \times \dfrac{8a}{10}$ When multiplying fractions, we multiply the numerators and the denominators. $z = \dfrac{ 2 \times 8a } { 8a \times 10}$ $z = \dfrac{16a}{80a}$ Simplify: $z = \dfrac{1}{5}$ |
343 | Simplify the following expression: $x = \dfrac{10}{5q - 1} \times \dfrac{3q}{7}$ | When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ 10 \times 3q } { (5q - 1) \times 7}$ $x = \dfrac{30q}{35q - 7}$ |
343 | Simplify the following expression: $p = \dfrac{9}{2n + 2} \div \dfrac{4}{10n}$ | Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{9}{2n + 2} \times \dfrac{10n}{4}$ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ 9 \times 10n } { (2n + 2) \times 4}$ $p = \dfrac{90n}{8n + 8}$ Simplify: $p = \dfrac{45n}{4n + 4}$ |
343 | Simplify the following expression: $r = \dfrac{8y + 6}{4} \times \dfrac{5}{10y}$ | When multiplying fractions, we multiply the numerators and the denominators. $r = \dfrac{ (8y + 6) \times 5 } { 4 \times 10y}$ $r = \dfrac{40y + 30}{40y}$ Simplify: $r = \dfrac{4y + 3}{4y}$ |
343 | Simplify the following expression: $y = \dfrac{6q - 8}{4} \times \dfrac{8}{10q}$ | When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ (6q - 8) \times 8 } { 4 \times 10q}$ $y = \dfrac{48q - 64}{40q}$ Simplify: $y = \dfrac{6q - 8}{5q}$ |
343 | Simplify the following expression: $r = \dfrac{8y + 6}{4} \div \dfrac{10y}{5}$ | Dividing by an expression is the same as multiplying by its inverse. $r = \dfrac{8y + 6}{4} \times \dfrac{5}{10y}$ When multiplying fractions, we multiply the numerators and the denominators. $r = \dfrac{ (8y + 6) \times 5 } { 4 \times 10y}$ $r = \dfrac{40y + 30}{40y}$ Simplify: $r = \dfrac{4y + 3}{4y}$ |
343 | Simplify the following expression: $q = \dfrac{y}{10} \div \dfrac{9y}{4}$ | Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{y}{10} \times \dfrac{4}{9y}$ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ y \times 4 } { 10 \times 9y}$ $q = \dfrac{4y}{90y}$ Simplify: $q = \dfrac{2}{45}$ |
343 | Simplify the following expression: $n = \dfrac{9}{5p} \div \dfrac{6}{2p}$ | Dividing by an expression is the same as multiplying by its inverse. $n = \dfrac{9}{5p} \times \dfrac{2p}{6}$ When multiplying fractions, we multiply the numerators and the denominators. $n = \dfrac{ 9 \times 2p } { 5p \times 6}$ $n = \dfrac{18p}{30p}$ Simplify: $n = \dfrac{3}{5}$ |
343 | Simplify the following expression: $y = \dfrac{5z + 1}{9} \div \dfrac{5z}{8}$ | Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{5z + 1}{9} \times \dfrac{8}{5z}$ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ (5z + 1) \times 8 } { 9 \times 5z}$ $y = \dfrac{40z + 8}{45z}$ |
343 | Simplify the following expression: $n = \dfrac{6}{3t} \div \dfrac{7}{7t}$ | Dividing by an expression is the same as multiplying by its inverse. $n = \dfrac{6}{3t} \times \dfrac{7t}{7}$ When multiplying fractions, we multiply the numerators and the denominators. $n = \dfrac{ 6 \times 7t } { 3t \times 7}$ $n = \dfrac{42t}{21t}$ Simplify: $n = \dfrac{2}{1}$ |
343 | Simplify the following expression: $y = \dfrac{3}{5k + 2} \times \dfrac{9k}{5}$ | When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ 3 \times 9k } { (5k + 2) \times 5}$ $y = \dfrac{27k}{25k + 10}$ |
343 | Simplify the following expression: $z = \dfrac{6}{5k + 5} \times \dfrac{3k}{6}$ | When multiplying fractions, we multiply the numerators and the denominators. $z = \dfrac{ 6 \times 3k } { (5k + 5) \times 6}$ $z = \dfrac{18k}{30k + 30}$ Simplify: $z = \dfrac{3k}{5k + 5}$ |
343 | Simplify the following expression: $a = \dfrac{7}{5k - 1} \times \dfrac{10k}{9}$ | When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ 7 \times 10k } { (5k - 1) \times 9}$ $a = \dfrac{70k}{45k - 9}$ |
343 | Simplify the following expression: $a = \dfrac{10}{x - 2} \div \dfrac{4}{8x}$ | Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{10}{x - 2} \times \dfrac{8x}{4}$ When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ 10 \times 8x } { (x - 2) \times 4}$ $a = \dfrac{80x}{4x - 8}$ Simplify: $a = \dfrac{20x}{x - 2}$ |
343 | Simplify the following expression: $n = \dfrac{10}{3x + 5} \div \dfrac{6}{5x}$ | Dividing by an expression is the same as multiplying by its inverse. $n = \dfrac{10}{3x + 5} \times \dfrac{5x}{6}$ When multiplying fractions, we multiply the numerators and the denominators. $n = \dfrac{ 10 \times 5x } { (3x + 5) \times 6}$ $n = \dfrac{50x}{18x + 30}$ Simplify: $n = \dfrac{25x}{9x + 15}$ |
343 | Simplify the following expression: $a = \dfrac{2n + 3}{6} \div \dfrac{2n}{6}$ | Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{2n + 3}{6} \times \dfrac{6}{2n}$ When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ (2n + 3) \times 6 } { 6 \times 2n}$ $a = \dfrac{12n + 18}{12n}$ Simplify: $a = \dfrac{2n + 3}{2n}$ |
343 | Simplify the following expression: $x = \dfrac{5}{6p - 10} \div \dfrac{7}{7p}$ | Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{5}{6p - 10} \times \dfrac{7p}{7}$ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ 5 \times 7p } { (6p - 10) \times 7}$ $x = \dfrac{35p}{42p - 70}$ Simplify: $x = \dfrac{5p}{6p - 10}$ |
343 | Simplify the following expression: $t = \dfrac{4r - 9}{7} \div \dfrac{7r}{10}$ | Dividing by an expression is the same as multiplying by its inverse. $t = \dfrac{4r - 9}{7} \times \dfrac{10}{7r}$ When multiplying fractions, we multiply the numerators and the denominators. $t = \dfrac{ (4r - 9) \times 10 } { 7 \times 7r}$ $t = \dfrac{40r - 90}{49r}$ |
343 | Simplify the following expression: $a = \dfrac{3q}{8} \div \dfrac{9q}{6}$ | Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{3q}{8} \times \dfrac{6}{9q}$ When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ 3q \times 6 } { 8 \times 9q}$ $a = \dfrac{18q}{72q}$ Simplify: $a = \dfrac{1}{4}$ |
343 | Simplify the following expression: $y = \dfrac{2p - 10}{9} \div \dfrac{3p}{2}$ | Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{2p - 10}{9} \times \dfrac{2}{3p}$ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ (2p - 10) \times 2 } { 9 \times 3p}$ $y = \dfrac{4p - 20}{27p}$ |
343 | Simplify the following expression: $z = \dfrac{2t}{4} \div \dfrac{10t}{6}$ | Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{2t}{4} \times \dfrac{6}{10t}$ When multiplying fractions, we multiply the numerators and the denominators. $z = \dfrac{ 2t \times 6 } { 4 \times 10t}$ $z = \dfrac{12t}{40t}$ Simplify: $z = \dfrac{3}{10}$ |
343 | Simplify the following expression: $q = \dfrac{z + 3}{4} \div \dfrac{9z}{5}$ | Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{z + 3}{4} \times \dfrac{5}{9z}$ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ (z + 3) \times 5 } { 4 \times 9z}$ $q = \dfrac{5z + 15}{36z}$ |
343 | Simplify the following expression: $k = \dfrac{5x - 9}{9} \times \dfrac{2}{6x}$ | When multiplying fractions, we multiply the numerators and the denominators. $k = \dfrac{ (5x - 9) \times 2 } { 9 \times 6x}$ $k = \dfrac{10x - 18}{54x}$ Simplify: $k = \dfrac{5x - 9}{27x}$ |
343 | Simplify the following expression: $z = \dfrac{3}{8t - 2} \div \dfrac{2}{10t}$ | Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{3}{8t - 2} \times \dfrac{10t}{2}$ When multiplying fractions, we multiply the numerators and the denominators. $z = \dfrac{ 3 \times 10t } { (8t - 2) \times 2}$ $z = \dfrac{30t}{16t - 4}$ Simplify: $z = \dfrac{15t}{8t - 2}$ |
343 | Simplify the following expression: $t = \dfrac{6}{5z - 2} \div \dfrac{2}{2z}$ | Dividing by an expression is the same as multiplying by its inverse. $t = \dfrac{6}{5z - 2} \times \dfrac{2z}{2}$ When multiplying fractions, we multiply the numerators and the denominators. $t = \dfrac{ 6 \times 2z } { (5z - 2) \times 2}$ $t = \dfrac{12z}{10z - 4}$ Simplify: $t = \dfrac{6z}{5z - 2}$ |
343 | Simplify the following expression: $t = \dfrac{8r}{4} \times \dfrac{3}{4r}$ | When multiplying fractions, we multiply the numerators and the denominators. $t = \dfrac{ 8r \times 3 } { 4 \times 4r}$ $t = \dfrac{24r}{16r}$ Simplify: $t = \dfrac{3}{2}$ |
343 | Simplify the following expression: $y = \dfrac{5}{8q + 10} \times \dfrac{7q}{7}$ | When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ 5 \times 7q } { (8q + 10) \times 7}$ $y = \dfrac{35q}{56q + 70}$ Simplify: $y = \dfrac{5q}{8q + 10}$ |
343 | Simplify the following expression: $a = \dfrac{7}{z - 10} \div \dfrac{3}{3z}$ | Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{7}{z - 10} \times \dfrac{3z}{3}$ When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ 7 \times 3z } { (z - 10) \times 3}$ $a = \dfrac{21z}{3z - 30}$ Simplify: $a = \dfrac{7z}{z - 10}$ |
343 | Simplify the following expression: $k = \dfrac{6r + 4}{5} \times \dfrac{3}{8r}$ | When multiplying fractions, we multiply the numerators and the denominators. $k = \dfrac{ (6r + 4) \times 3 } { 5 \times 8r}$ $k = \dfrac{18r + 12}{40r}$ Simplify: $k = \dfrac{9r + 6}{20r}$ |
343 | Simplify the following expression: $a = \dfrac{z - 9}{8} \times \dfrac{7}{5z}$ | When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ (z - 9) \times 7 } { 8 \times 5z}$ $a = \dfrac{7z - 63}{40z}$ |
343 | Simplify the following expression: $n = \dfrac{6}{3t} \times \dfrac{7t}{7}$ | When multiplying fractions, we multiply the numerators and the denominators. $n = \dfrac{ 6 \times 7t } { 3t \times 7}$ $n = \dfrac{42t}{21t}$ Simplify: $n = \dfrac{2}{1}$ |
185 | Let $f(x) = -2x^{2}+2x+9$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-2x^{2}+2x+9 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -2, b = 2, c = 9$ $ x = \dfrac{-2 \pm \sqrt{2^{2} - 4 \cdot -2 \cdot 9}}{2 \cdot -2}$ $ x = \dfrac{-2 \pm \sqrt{76}}{-4}$ $ x = \dfrac{-2 \pm 2\sqrt{19}}{-4}$ $x =\dfrac{-1 \pm \sqrt{19}}{-2}$ |
185 | Let $f(x) = 5x^{2}+10x-2$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $5x^{2}+10x-2 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 5, b = 10, c = -2$ $ x = \dfrac{-10 \pm \sqrt{10^{2} - 4 \cdot 5 \cdot -2}}{2 \cdot 5}$ $ x = \dfrac{-10 \pm \sqrt{140}}{10}$ $ x = \dfrac{-10 \pm 2\sqrt{35}}{10}$ $x =\dfrac{-5 \pm \sqrt{35}}{5}$ |
185 | Let $f(x) = x^{2}-9x+3$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $x^{2}-9x+3 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 1, b = -9, c = 3$ $ x = \dfrac{+ 9 \pm \sqrt{(-9)^{2} - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$ $ x = \dfrac{9 \pm \sqrt{69}}{2}$ $ x = \dfrac{9 \pm \sqrt{69}}{2}$ $x =\dfrac{9 \pm \sqrt{69}}{2}$ |
185 | Let $f(x) = -x^{2}-10x+1$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-x^{2}-10x+1 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -1, b = -10, c = 1$ $ x = \dfrac{+ 10 \pm \sqrt{(-10)^{2} - 4 \cdot -1 \cdot 1}}{2 \cdot -1}$ $ x = \dfrac{10 \pm \sqrt{104}}{-2}$ $ x = \dfrac{10 \pm 2\sqrt{26}}{-2}$ $x =-5 \pm \sqrt{26}$ |
185 | Let $f(x) = -9x^{2}+9x+2$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-9x^{2}+9x+2 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -9, b = 9, c = 2$ $ x = \dfrac{-9 \pm \sqrt{9^{2} - 4 \cdot -9 \cdot 2}}{2 \cdot -9}$ $ x = \dfrac{-9 \pm \sqrt{153}}{-18}$ $ x = \dfrac{-9 \pm 3\sqrt{17}}{-18}$ $x =\dfrac{-3 \pm \sqrt{17}}{-6}$ |
185 | Let $f(x) = -6x^{2}+6x+6$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-6x^{2}+6x+6 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -6, b = 6, c = 6$ $ x = \dfrac{-6 \pm \sqrt{6^{2} - 4 \cdot -6 \cdot 6}}{2 \cdot -6}$ $ x = \dfrac{-6 \pm \sqrt{180}}{-12}$ $ x = \dfrac{-6 \pm 6\sqrt{5}}{-12}$ $x =\dfrac{-1 \pm \sqrt{5}}{-2}$ |
185 | Let $f(x) = -8x^{2}-4x+1$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-8x^{2}-4x+1 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -8, b = -4, c = 1$ $ x = \dfrac{+ 4 \pm \sqrt{(-4)^{2} - 4 \cdot -8 \cdot 1}}{2 \cdot -8}$ $ x = \dfrac{4 \pm \sqrt{48}}{-16}$ $ x = \dfrac{4 \pm 4\sqrt{3}}{-16}$ $x =\dfrac{1 \pm \sqrt{3}}{-4}$ |
185 | Let $f(x) = -x^{2}+5x-5$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-x^{2}+5x-5 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -1, b = 5, c = -5$ $ x = \dfrac{-5 \pm \sqrt{5^{2} - 4 \cdot -1 \cdot -5}}{2 \cdot -1}$ $ x = \dfrac{-5 \pm \sqrt{5}}{-2}$ $ x = \dfrac{-5 \pm \sqrt{5}}{-2}$ $x =\dfrac{-5 \pm \sqrt{5}}{-2}$ |
185 | Let $f(x) = 8x^{2}+8x-1$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $8x^{2}+8x-1 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 8, b = 8, c = -1$ $ x = \dfrac{-8 \pm \sqrt{8^{2} - 4 \cdot 8 \cdot -1}}{2 \cdot 8}$ $ x = \dfrac{-8 \pm \sqrt{96}}{16}$ $ x = \dfrac{-8 \pm 4\sqrt{6}}{16}$ $x =\dfrac{-2 \pm \sqrt{6}}{4}$ |
185 | Let $f(x) = -2x^{2}-3x+3$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-2x^{2}-3x+3 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -2, b = -3, c = 3$ $ x = \dfrac{+ 3 \pm \sqrt{(-3)^{2} - 4 \cdot -2 \cdot 3}}{2 \cdot -2}$ $ x = \dfrac{3 \pm \sqrt{33}}{-4}$ $ x = \dfrac{3 \pm \sqrt{33}}{-4}$ $x =\dfrac{3 \pm \sqrt{33}}{-4}$ |
185 | Let $f(x) = -6x^{2}+5x-1$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-6x^{2}+5x-1 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -6, b = 5, c = -1$ $ x = \dfrac{-5 \pm \sqrt{5^{2} - 4 \cdot -6 \cdot -1}}{2 \cdot -6}$ $ x = \dfrac{-5 \pm \sqrt{1}}{-12}$ $ x = \dfrac{-5 \pm 1}{-12}$ $x =\dfrac{-5 \pm \sqrt{1}}{-12}$ |
185 | Let $f(x) = -5x^{2}+5x+1$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-5x^{2}+5x+1 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -5, b = 5, c = 1$ $ x = \dfrac{-5 \pm \sqrt{5^{2} - 4 \cdot -5 \cdot 1}}{2 \cdot -5}$ $ x = \dfrac{-5 \pm \sqrt{45}}{-10}$ $ x = \dfrac{-5 \pm 3\sqrt{5}}{-10}$ $x =\dfrac{-5 \pm 3\sqrt{5}}{-10}$ |
185 | Let $f(x) = -6x^{2}+5x+1$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-6x^{2}+5x+1 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -6, b = 5, c = 1$ $ x = \dfrac{-5 \pm \sqrt{5^{2} - 4 \cdot -6 \cdot 1}}{2 \cdot -6}$ $ x = \dfrac{-5 \pm \sqrt{49}}{-12}$ $ x = \dfrac{-5 \pm 7}{-12}$ $x =-\frac{1}{6},1$ |
185 | Let $f(x) = x^{2}-6x-5$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $x^{2}-6x-5 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 1, b = -6, c = -5$ $ x = \dfrac{+ 6 \pm \sqrt{(-6)^{2} - 4 \cdot 1 \cdot -5}}{2 \cdot 1}$ $ x = \dfrac{6 \pm \sqrt{56}}{2}$ $ x = \dfrac{6 \pm 2\sqrt{14}}{2}$ $x =3 \pm \sqrt{14}$ |
185 | Let $f(x) = -6x^{2}+10x+3$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-6x^{2}+10x+3 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -6, b = 10, c = 3$ $ x = \dfrac{-10 \pm \sqrt{10^{2} - 4 \cdot -6 \cdot 3}}{2 \cdot -6}$ $ x = \dfrac{-10 \pm \sqrt{172}}{-12}$ $ x = \dfrac{-10 \pm 2\sqrt{43}}{-12}$ $x =\dfrac{-5 \pm \sqrt{43}}{-6}$ |
185 | Let $f(x) = 7x^{2}+10x+2$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $7x^{2}+10x+2 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 7, b = 10, c = 2$ $ x = \dfrac{-10 \pm \sqrt{10^{2} - 4 \cdot 7 \cdot 2}}{2 \cdot 7}$ $ x = \dfrac{-10 \pm \sqrt{44}}{14}$ $ x = \dfrac{-10 \pm 2\sqrt{11}}{14}$ $x =\dfrac{-5 \pm \sqrt{11}}{7}$ |
185 | Let $f(x) = -4x^{2}+9x+10$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-4x^{2}+9x+10 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -4, b = 9, c = 10$ $ x = \dfrac{-9 \pm \sqrt{9^{2} - 4 \cdot -4 \cdot 10}}{2 \cdot -4}$ $ x = \dfrac{-9 \pm \sqrt{241}}{-8}$ $ x = \dfrac{-9 \pm \sqrt{241}}{-8}$ $x =\dfrac{-9 \pm \sqrt{241}}{-8}$ |
185 | Let $f(x) = -3x^{2}+8x+9$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-3x^{2}+8x+9 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -3, b = 8, c = 9$ $ x = \dfrac{-8 \pm \sqrt{8^{2} - 4 \cdot -3 \cdot 9}}{2 \cdot -3}$ $ x = \dfrac{-8 \pm \sqrt{172}}{-6}$ $ x = \dfrac{-8 \pm 2\sqrt{43}}{-6}$ $x =\dfrac{-4 \pm \sqrt{43}}{-3}$ |
185 | Let $f(x) = 6x^{2}+10x-8$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $6x^{2}+10x-8 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 6, b = 10, c = -8$ $ x = \dfrac{-10 \pm \sqrt{10^{2} - 4 \cdot 6 \cdot -8}}{2 \cdot 6}$ $ x = \dfrac{-10 \pm \sqrt{292}}{12}$ $ x = \dfrac{-10 \pm 2\sqrt{73}}{12}$ $x =\dfrac{-5 \pm \sqrt{73}}{6}$ |
185 | Let $f(x) = 8x^{2}+x-5$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $8x^{2}+x-5 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 8, b = 1, c = -5$ $ x = \dfrac{-1 \pm \sqrt{1^{2} - 4 \cdot 8 \cdot -5}}{2 \cdot 8}$ $ x = \dfrac{-1 \pm \sqrt{161}}{16}$ $ x = \dfrac{-1 \pm \sqrt{161}}{16}$ $x =\dfrac{-1 \pm \sqrt{161}}{16}$ |
185 | Let $f(x) = -7x^{2}+8x+6$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-7x^{2}+8x+6 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -7, b = 8, c = 6$ $ x = \dfrac{-8 \pm \sqrt{8^{2} - 4 \cdot -7 \cdot 6}}{2 \cdot -7}$ $ x = \dfrac{-8 \pm \sqrt{232}}{-14}$ $ x = \dfrac{-8 \pm 2\sqrt{58}}{-14}$ $x =\dfrac{-4 \pm \sqrt{58}}{-7}$ |
185 | Let $f(x) = 3x^{2}+8x+5$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $3x^{2}+8x+5 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 3, b = 8, c = 5$ $ x = \dfrac{-8 \pm \sqrt{8^{2} - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$ $ x = \dfrac{-8 \pm \sqrt{4}}{6}$ $ x = \dfrac{-8 \pm 2}{6}$ $x =-1,-\frac{5}{3}$ |
185 | Let $f(x) = 7x^{2}+8x-6$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $7x^{2}+8x-6 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 7, b = 8, c = -6$ $ x = \dfrac{-8 \pm \sqrt{8^{2} - 4 \cdot 7 \cdot -6}}{2 \cdot 7}$ $ x = \dfrac{-8 \pm \sqrt{232}}{14}$ $ x = \dfrac{-8 \pm 2\sqrt{58}}{14}$ $x =\dfrac{-4 \pm \sqrt{58}}{7}$ |
185 | Let $f(x) = 3x^{2}-8x-8$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $3x^{2}-8x-8 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 3, b = -8, c = -8$ $ x = \dfrac{+ 8 \pm \sqrt{(-8)^{2} - 4 \cdot 3 \cdot -8}}{2 \cdot 3}$ $ x = \dfrac{8 \pm \sqrt{160}}{6}$ $ x = \dfrac{8 \pm 4\sqrt{10}}{6}$ $x =\dfrac{4 \pm 2\sqrt{10}}{3}$ |
185 | Let $f(x) = -8x^{2}-8x+4$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-8x^{2}-8x+4 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -8, b = -8, c = 4$ $ x = \dfrac{+ 8 \pm \sqrt{(-8)^{2} - 4 \cdot -8 \cdot 4}}{2 \cdot -8}$ $ x = \dfrac{8 \pm \sqrt{192}}{-16}$ $ x = \dfrac{8 \pm 8\sqrt{3}}{-16}$ $x =\dfrac{1 \pm \sqrt{3}}{-2}$ |
185 | Let $f(x) = -5x^{2}+6x+2$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-5x^{2}+6x+2 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -5, b = 6, c = 2$ $ x = \dfrac{-6 \pm \sqrt{6^{2} - 4 \cdot -5 \cdot 2}}{2 \cdot -5}$ $ x = \dfrac{-6 \pm \sqrt{76}}{-10}$ $ x = \dfrac{-6 \pm 2\sqrt{19}}{-10}$ $x =\dfrac{-3 \pm \sqrt{19}}{-5}$ |
185 | Let $f(x) = x^{2}-4x-8$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $x^{2}-4x-8 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 1, b = -4, c = -8$ $ x = \dfrac{+ 4 \pm \sqrt{(-4)^{2} - 4 \cdot 1 \cdot -8}}{2 \cdot 1}$ $ x = \dfrac{4 \pm \sqrt{48}}{2}$ $ x = \dfrac{4 \pm 4\sqrt{3}}{2}$ $x =2 \pm 2\sqrt{3}$ |
185 | Let $f(x) = -10x^{2}+10x-1$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-10x^{2}+10x-1 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -10, b = 10, c = -1$ $ x = \dfrac{-10 \pm \sqrt{10^{2} - 4 \cdot -10 \cdot -1}}{2 \cdot -10}$ $ x = \dfrac{-10 \pm \sqrt{60}}{-20}$ $ x = \dfrac{-10 \pm 2\sqrt{15}}{-20}$ $x =\dfrac{-5 \pm \sqrt{15}}{-10}$ |
185 | Let $f(x) = -5x^{2}-2x+6$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-5x^{2}-2x+6 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -5, b = -2, c = 6$ $ x = \dfrac{+ 2 \pm \sqrt{(-2)^{2} - 4 \cdot -5 \cdot 6}}{2 \cdot -5}$ $ x = \dfrac{2 \pm \sqrt{124}}{-10}$ $ x = \dfrac{2 \pm 2\sqrt{31}}{-10}$ $x =\dfrac{1 \pm \sqrt{31}}{-5}$ |
185 | Let $f(x) = 3x^{2}+9x+3$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $3x^{2}+9x+3 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 3, b = 9, c = 3$ $ x = \dfrac{-9 \pm \sqrt{9^{2} - 4 \cdot 3 \cdot 3}}{2 \cdot 3}$ $ x = \dfrac{-9 \pm \sqrt{45}}{6}$ $ x = \dfrac{-9 \pm 3\sqrt{5}}{6}$ $x =\dfrac{-3 \pm \sqrt{5}}{2}$ |
185 | Let $f(x) = 6x^{2}-9x-4$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $6x^{2}-9x-4 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 6, b = -9, c = -4$ $ x = \dfrac{+ 9 \pm \sqrt{(-9)^{2} - 4 \cdot 6 \cdot -4}}{2 \cdot 6}$ $ x = \dfrac{9 \pm \sqrt{177}}{12}$ $ x = \dfrac{9 \pm \sqrt{177}}{12}$ $x =\dfrac{9 \pm \sqrt{177}}{12}$ |
185 | Let $f(x) = 2x^{2}+10x+8$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $2x^{2}+10x+8 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 2, b = 10, c = 8$ $ x = \dfrac{-10 \pm \sqrt{10^{2} - 4 \cdot 2 \cdot 8}}{2 \cdot 2}$ $ x = \dfrac{-10 \pm \sqrt{36}}{4}$ $ x = \dfrac{-10 \pm 6}{4}$ $x =-1,-4$ |
185 | Let $f(x) = -2x^{2}-x+7$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-2x^{2}-x+7 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -2, b = -1, c = 7$ $ x = \dfrac{+ 1 \pm \sqrt{(-1)^{2} - 4 \cdot -2 \cdot 7}}{2 \cdot -2}$ $ x = \dfrac{1 \pm \sqrt{57}}{-4}$ $ x = \dfrac{1 \pm \sqrt{57}}{-4}$ $x =\dfrac{1 \pm \sqrt{57}}{-4}$ |
185 | Let $f(x) = 10x^{2}+4x-8$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $10x^{2}+4x-8 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 10, b = 4, c = -8$ $ x = \dfrac{-4 \pm \sqrt{4^{2} - 4 \cdot 10 \cdot -8}}{2 \cdot 10}$ $ x = \dfrac{-4 \pm \sqrt{336}}{20}$ $ x = \dfrac{-4 \pm 4\sqrt{21}}{20}$ $x =\dfrac{-1 \pm \sqrt{21}}{5}$ |
185 | Let $f(x) = -7x^{2}+x+9$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-7x^{2}+x+9 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -7, b = 1, c = 9$ $ x = \dfrac{-1 \pm \sqrt{1^{2} - 4 \cdot -7 \cdot 9}}{2 \cdot -7}$ $ x = \dfrac{-1 \pm \sqrt{253}}{-14}$ $ x = \dfrac{-1 \pm \sqrt{253}}{-14}$ $x =\dfrac{-1 \pm \sqrt{253}}{-14}$ |
185 | Let $f(x) = -9x^{2}+9x+9$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-9x^{2}+9x+9 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -9, b = 9, c = 9$ $ x = \dfrac{-9 \pm \sqrt{9^{2} - 4 \cdot -9 \cdot 9}}{2 \cdot -9}$ $ x = \dfrac{-9 \pm \sqrt{405}}{-18}$ $ x = \dfrac{-9 \pm 9\sqrt{5}}{-18}$ $x =\dfrac{-1 \pm \sqrt{5}}{-2}$ |
185 | Let $f(x) = -x^{2}+10x+4$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-x^{2}+10x+4 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -1, b = 10, c = 4$ $ x = \dfrac{-10 \pm \sqrt{10^{2} - 4 \cdot -1 \cdot 4}}{2 \cdot -1}$ $ x = \dfrac{-10 \pm \sqrt{116}}{-2}$ $ x = \dfrac{-10 \pm 2\sqrt{29}}{-2}$ $x =5 \pm \sqrt{29}$ |
185 | Let $f(x) = -9x^{2}+x+10$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-9x^{2}+x+10 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -9, b = 1, c = 10$ $ x = \dfrac{-1 \pm \sqrt{1^{2} - 4 \cdot -9 \cdot 10}}{2 \cdot -9}$ $ x = \dfrac{-1 \pm \sqrt{361}}{-18}$ $ x = \dfrac{-1 \pm 19}{-18}$ $x =-1,\frac{10}{9}$ |
185 | Let $f(x) = -2x^{2}-10x-8$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-2x^{2}-10x-8 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -2, b = -10, c = -8$ $ x = \dfrac{+ 10 \pm \sqrt{(-10)^{2} - 4 \cdot -2 \cdot -8}}{2 \cdot -2}$ $ x = \dfrac{10 \pm \sqrt{36}}{-4}$ $ x = \dfrac{10 \pm 6}{-4}$ $x =-4,-1$ |
185 | Let $f(x) = -6x^{2}-3x+6$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-6x^{2}-3x+6 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -6, b = -3, c = 6$ $ x = \dfrac{+ 3 \pm \sqrt{(-3)^{2} - 4 \cdot -6 \cdot 6}}{2 \cdot -6}$ $ x = \dfrac{3 \pm \sqrt{153}}{-12}$ $ x = \dfrac{3 \pm 3\sqrt{17}}{-12}$ $x =\dfrac{1 \pm \sqrt{17}}{-4}$ |
185 | Let $f(x) = 9x^{2}-9x-7$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $9x^{2}-9x-7 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 9, b = -9, c = -7$ $ x = \dfrac{+ 9 \pm \sqrt{(-9)^{2} - 4 \cdot 9 \cdot -7}}{2 \cdot 9}$ $ x = \dfrac{9 \pm \sqrt{333}}{18}$ $ x = \dfrac{9 \pm 3\sqrt{37}}{18}$ $x =\dfrac{3 \pm \sqrt{37}}{6}$ |
185 | Let $f(x) = -10x^{2}-7x+8$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-10x^{2}-7x+8 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -10, b = -7, c = 8$ $ x = \dfrac{+ 7 \pm \sqrt{(-7)^{2} - 4 \cdot -10 \cdot 8}}{2 \cdot -10}$ $ x = \dfrac{7 \pm \sqrt{369}}{-20}$ $ x = \dfrac{7 \pm 3\sqrt{41}}{-20}$ $x =\dfrac{7 \pm 3\sqrt{41}}{-20}$ |
185 | Let $f(x) = -2x^{2}-5x-2$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-2x^{2}-5x-2 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -2, b = -5, c = -2$ $ x = \dfrac{+ 5 \pm \sqrt{(-5)^{2} - 4 \cdot -2 \cdot -2}}{2 \cdot -2}$ $ x = \dfrac{5 \pm \sqrt{9}}{-4}$ $ x = \dfrac{5 \pm 3}{-4}$ $x =-2,-\frac{1}{2}$ |
185 | Let $f(x) = 5x^{2}+x-8$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $5x^{2}+x-8 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 5, b = 1, c = -8$ $ x = \dfrac{-1 \pm \sqrt{1^{2} - 4 \cdot 5 \cdot -8}}{2 \cdot 5}$ $ x = \dfrac{-1 \pm \sqrt{161}}{10}$ $ x = \dfrac{-1 \pm \sqrt{161}}{10}$ $x =\dfrac{-1 \pm \sqrt{161}}{10}$ |
185 | Let $f(x) = -8x^{2}+8x+6$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-8x^{2}+8x+6 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -8, b = 8, c = 6$ $ x = \dfrac{-8 \pm \sqrt{8^{2} - 4 \cdot -8 \cdot 6}}{2 \cdot -8}$ $ x = \dfrac{-8 \pm \sqrt{256}}{-16}$ $ x = \dfrac{-8 \pm 16}{-16}$ $x =-\frac{1}{2},\frac{3}{2}$ |
185 | Let $f(x) = -6x^{2}-4x+6$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-6x^{2}-4x+6 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -6, b = -4, c = 6$ $ x = \dfrac{+ 4 \pm \sqrt{(-4)^{2} - 4 \cdot -6 \cdot 6}}{2 \cdot -6}$ $ x = \dfrac{4 \pm \sqrt{160}}{-12}$ $ x = \dfrac{4 \pm 4\sqrt{10}}{-12}$ $x =\dfrac{1 \pm \sqrt{10}}{-3}$ |
185 | Let $f(x) = -9x^{2}+3x+5$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-9x^{2}+3x+5 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -9, b = 3, c = 5$ $ x = \dfrac{-3 \pm \sqrt{3^{2} - 4 \cdot -9 \cdot 5}}{2 \cdot -9}$ $ x = \dfrac{-3 \pm \sqrt{189}}{-18}$ $ x = \dfrac{-3 \pm 3\sqrt{21}}{-18}$ $x =\dfrac{-1 \pm \sqrt{21}}{-6}$ |
185 | Let $f(x) = -7x^{2}+8x+10$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-7x^{2}+8x+10 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -7, b = 8, c = 10$ $ x = \dfrac{-8 \pm \sqrt{8^{2} - 4 \cdot -7 \cdot 10}}{2 \cdot -7}$ $ x = \dfrac{-8 \pm \sqrt{344}}{-14}$ $ x = \dfrac{-8 \pm 2\sqrt{86}}{-14}$ $x =\dfrac{-4 \pm \sqrt{86}}{-7}$ |
185 | Let $f(x) = -8x^{2}+10x-3$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-8x^{2}+10x-3 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -8, b = 10, c = -3$ $ x = \dfrac{-10 \pm \sqrt{10^{2} - 4 \cdot -8 \cdot -3}}{2 \cdot -8}$ $ x = \dfrac{-10 \pm \sqrt{4}}{-16}$ $ x = \dfrac{-10 \pm 2}{-16}$ $x =\frac{1}{2},\frac{3}{4}$ |
185 | Let $f(x) = -3x^{2}+x+10$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-3x^{2}+x+10 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -3, b = 1, c = 10$ $ x = \dfrac{-1 \pm \sqrt{1^{2} - 4 \cdot -3 \cdot 10}}{2 \cdot -3}$ $ x = \dfrac{-1 \pm \sqrt{121}}{-6}$ $ x = \dfrac{-1 \pm 11}{-6}$ $x =-\frac{5}{3},2$ |
185 | Let $f(x) = -3x^{2}-6x-1$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-3x^{2}-6x-1 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -3, b = -6, c = -1$ $ x = \dfrac{+ 6 \pm \sqrt{(-6)^{2} - 4 \cdot -3 \cdot -1}}{2 \cdot -3}$ $ x = \dfrac{6 \pm \sqrt{24}}{-6}$ $ x = \dfrac{6 \pm 2\sqrt{6}}{-6}$ $x =\dfrac{3 \pm \sqrt{6}}{-3}$ |
185 | Let $f(x) = 10x^{2}-9x-6$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $10x^{2}-9x-6 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 10, b = -9, c = -6$ $ x = \dfrac{+ 9 \pm \sqrt{(-9)^{2} - 4 \cdot 10 \cdot -6}}{2 \cdot 10}$ $ x = \dfrac{9 \pm \sqrt{321}}{20}$ $ x = \dfrac{9 \pm \sqrt{321}}{20}$ $x =\dfrac{9 \pm \sqrt{321}}{20}$ |
185 | Let $f(x) = -3x^{2}-8x+4$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-3x^{2}-8x+4 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -3, b = -8, c = 4$ $ x = \dfrac{+ 8 \pm \sqrt{(-8)^{2} - 4 \cdot -3 \cdot 4}}{2 \cdot -3}$ $ x = \dfrac{8 \pm \sqrt{112}}{-6}$ $ x = \dfrac{8 \pm 4\sqrt{7}}{-6}$ $x =\dfrac{4 \pm 2\sqrt{7}}{-3}$ |
185 | Let $f(x) = -4x^{2}-10x+5$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-4x^{2}-10x+5 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -4, b = -10, c = 5$ $ x = \dfrac{+ 10 \pm \sqrt{(-10)^{2} - 4 \cdot -4 \cdot 5}}{2 \cdot -4}$ $ x = \dfrac{10 \pm \sqrt{180}}{-8}$ $ x = \dfrac{10 \pm 6\sqrt{5}}{-8}$ $x =\dfrac{5 \pm 3\sqrt{5}}{-4}$ |
185 | Let $f(x) = -7x^{2}+4x+7$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )? | The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-7x^{2}+4x+7 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -7, b = 4, c = 7$ $ x = \dfrac{-4 \pm \sqrt{4^{2} - 4 \cdot -7 \cdot 7}}{2 \cdot -7}$ $ x = \dfrac{-4 \pm \sqrt{212}}{-14}$ $ x = \dfrac{-4 \pm 2\sqrt{53}}{-14}$ $x =\dfrac{-2 \pm \sqrt{53}}{-7}$ |
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