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Solve the quadratic equation
Answer
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Number of solution
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Relationship between roots and coefficients
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$y = 2 x ^ { 2 } - x - 15$
$y = 0$
$x$Intercept
$\left ( 3 , 0 \right )$, $\left ( - \dfrac { 5 } { 2 } , 0 \right )$
$y$Intercept
$\left ( 0 , - 15 \right )$
$\begin{array} {l} x = 3 \\ x = - \dfrac { 5 } { 2 } \end{array}$
Find solution by method of factorization
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 15 } = 0$
$acx^{2} + \left(ad + bc\right)x +bd = \left(ax + b\right)\left(cx+d\right)$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 5 } \right ) = 0$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 5 } \right ) = \color{#FF6800}{ 0 }$
$ $ If the product of the factor is 0, at least one factor should be 0 $ $
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 5 } = \color{#FF6800}{ 0 } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 5 } = \color{#FF6800}{ 0 } \end{array}$
$ $ Solve the equation to find $ x$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 3 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 5 } } { \color{#FF6800}{ 2 } } } \end{array}$
$\begin{array} {l} x = 3 \\ x = - \dfrac { 5 } { 2 } \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 15 } = \color{#FF6800}{ 0 }$
$ $ Divide both sides by the coefficient of the leading highest term $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 15 } } { \color{#FF6800}{ 2 } } } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 15 } } { \color{#FF6800}{ 2 } } } = \color{#FF6800}{ 0 }$
$ $ Convert the quadratic expression on the left side to a perfect square format $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 15 } } { \color{#FF6800}{ 2 } } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( x - \dfrac { 1 } { 4 } \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 15 } } { \color{#FF6800}{ 2 } } } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
$ $ Move the constant to the right side and change the sign $ $
$\left ( x - \dfrac { 1 } { 4 } \right ) ^ { 2 } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 15 } } { \color{#FF6800}{ 2 } } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \right ) ^ { \color{#FF6800}{ 2 } }$
$\left ( x - \dfrac { 1 } { 4 } \right ) ^ { 2 } = \dfrac { 15 } { 2 } + \left ( \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \right ) ^ { \color{#FF6800}{ 2 } }$
$ $ When raising a fraction to the power, raise the numerator and denominator each to the power $ $
$\left ( x - \dfrac { 1 } { 4 } \right ) ^ { 2 } = \dfrac { 15 } { 2 } + \dfrac { \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 15 } } { \color{#FF6800}{ 2 } } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } } }$
$ $ Organize the expression $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 121 } } { \color{#FF6800}{ 16 } } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 121 } } { \color{#FF6800}{ 16 } } }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { \color{#FF6800}{ 121 } } { \color{#FF6800}{ 16 } } } }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { \color{#FF6800}{ 121 } } { \color{#FF6800}{ 16 } } } }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { \color{#FF6800}{ 11 } } { \color{#FF6800}{ 4 } } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { \color{#FF6800}{ 11 } } { \color{#FF6800}{ 4 } } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 11 } } { \color{#FF6800}{ 4 } } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 11 } } { \color{#FF6800}{ 4 } } } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 11 } } { \color{#FF6800}{ 4 } } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 11 } } { \color{#FF6800}{ 4 } } } \end{array}$
$ $ Organize the expression $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 3 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 5 } } { \color{#FF6800}{ 2 } } } \end{array}$
$\begin{array} {l} x = 3 \\ x = - \dfrac { 5 } { 2 } \end{array}$
Calculate using the quadratic formula
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 15 } = \color{#FF6800}{ 0 }$
$ $ Solve the quadratic equation $ ax^{2}+bx+c=0 $ using the quadratic formula $ \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 15 } \right ) } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } }$
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 1 \right ) \pm \sqrt{ \left ( - 1 \right ) ^ { 2 } - 4 \times 2 \times \left ( - 15 \right ) } } { 2 \times 2 }$
$ $ Simplify Minus $ $
$x = \dfrac { 1 \pm \sqrt{ \left ( - 1 \right ) ^ { 2 } - 4 \times 2 \times \left ( - 15 \right ) } } { 2 \times 2 }$
$x = \dfrac { 1 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times \left ( - 15 \right ) } } { 2 \times 2 }$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$x = \dfrac { 1 \pm \sqrt{ 1 ^ { 2 } - 4 \times 2 \times \left ( - 15 \right ) } } { 2 \times 2 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } \pm \sqrt{ \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 15 } \right ) } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } \pm \sqrt{ \color{#FF6800}{ 121 } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } }$
$x = \dfrac { 1 \pm \sqrt{ \color{#FF6800}{ 121 } } } { 2 \times 2 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { 1 \pm \color{#FF6800}{ 11 } } { 2 \times 2 }$
$x = \dfrac { 1 \pm 11 } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } }$
$ $ Multiply $ 2 $ and $ 2$
$x = \dfrac { 1 \pm 11 } { \color{#FF6800}{ 4 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } \pm \color{#FF6800}{ 11 } } { \color{#FF6800}{ 4 } } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 11 } } { \color{#FF6800}{ 4 } } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 11 } } { \color{#FF6800}{ 4 } } } \end{array}$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 11 } } { 4 } \\ x = \dfrac { 1 - 11 } { 4 } \end{array}$
$ $ Add $ 1 $ and $ 11$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 12 } } { 4 } \\ x = \dfrac { 1 - 11 } { 4 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 12 } } { \color{#FF6800}{ 4 } } } \\ x = \dfrac { 1 - 11 } { 4 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } } { \color{#FF6800}{ 1 } } } \\ x = \dfrac { 1 - 11 } { 4 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } } { \color{#FF6800}{ 1 } } } \\ x = \dfrac { 1 - 11 } { 4 } \end{array}$
$ $ Reduce the fraction to the lowest term $ $
$\begin{array} {l} x = \color{#FF6800}{ 3 } \\ x = \dfrac { 1 - 11 } { 4 } \end{array}$
$\begin{array} {l} x = 3 \\ x = \dfrac { \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 11 } } { 4 } \end{array}$
$ $ Subtract $ 11 $ from $ 1$
$\begin{array} {l} x = 3 \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 10 } } { 4 } \end{array}$
$\begin{array} {l} x = 3 \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 10 } } { \color{#FF6800}{ 4 } } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = 3 \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 5 } } { \color{#FF6800}{ 2 } } } \end{array}$
$\begin{array} {l} x = 3 \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 5 } } { 2 } \end{array}$
$ $ Move the minus sign to the front of the fraction $ $
$\begin{array} {l} x = 3 \\ x = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 5 } } { \color{#FF6800}{ 2 } } } \end{array}$
$ $ 2 real roots $ $
Find the number of solutions
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 15 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 15 } \right )$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times \left ( - 15 \right )$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$D = 1 ^ { 2 } - 4 \times 2 \times \left ( - 15 \right )$
$D = \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times \left ( - 15 \right )$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 1 } - 4 \times 2 \times \left ( - 15 \right )$
$D = 1 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 15 } \right )$
$ $ Multiply the numbers $ $
$D = 1 + \color{#FF6800}{ 120 }$
$D = \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 120 }$
$ $ Add $ 1 $ and $ 120$
$D = \color{#FF6800}{ 121 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 121 }$
$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $
$ $ 2 real roots $ $
$\alpha + \beta = \dfrac { 1 } { 2 } , \alpha \beta = - \dfrac { 15 } { 2 }$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 15 } = \color{#FF6800}{ 0 }$
$ $ In the quadratic equation $ ax^{2}+bx+c=0 $ , if the two roots are $ \alpha, \beta $ , then it is $ \alpha + \beta =-\dfrac{b}{a} $ , $ \alpha\times\beta=\dfrac{c}{a}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 15 } } { \color{#FF6800}{ 2 } } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } , \alpha \beta = \dfrac { - 15 } { 2 }$
$ $ Solve the sign of a fraction with a negative sign $ $
$\alpha + \beta = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } , \alpha \beta = \dfrac { - 15 } { 2 }$
$\alpha + \beta = \dfrac { 1 } { 2 } , \alpha \beta = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 15 } } { 2 }$
$ $ Move the minus sign to the front of the fraction $ $
$\alpha + \beta = \dfrac { 1 } { 2 } , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 15 } } { \color{#FF6800}{ 2 } } }$
Solution search results
search-thumbnail-2) Solve: Show work for full credit. Simplify solutic
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$∠$ $F$ From the sum $ot$ $x^{A}\left(2\right)-8$ with $3x-12$ subtract the sum $ofx-9$ with $3x-x^{A}\left(2\right)$ Multinlication takes place $\left(2\times A$ $\left(2\right)+5x+21\right)\left(x-$ $5\right)=$ #Please provide solution by using math formula.
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search-thumbnail-2\times=15=0
$2\times =15=0$
10th-13th grade
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search-thumbnail-8x^{2} -2x-15 =0
$8x^{2}$ $-2x-15$ $=0$
7th-9th grade
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search-thumbnail-8 8 What is the product of the roots of quadratic equation 5x^{2}-x-15=07
$8$ 8 What is the product of the roots of quadratic equation $5x^{2}-x-15=07$ $8$ $9$ b. $7$ $c-5$ $2$ $40$ $7$ $∩$
7th-9th grade
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Search count: 106
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search-thumbnail-8x^{2}-2x- 15 = 0
$8x^{2}-2x-$ $15$ $=$ $0$
7th-9th grade
Other
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