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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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Graph
$y = \left ( 2 x + 3 \right ) ^ { 2 } - 4 \left ( 2 x + 3 \right ) - 32$
$y = 0$
$x$Intercept
$\left ( \dfrac { 5 } { 2 } , 0 \right )$, $\left ( - \dfrac { 7 } { 2 } , 0 \right )$
$y$Intercept
$\left ( 0 , - 35 \right )$
Minimum
$\left ( - \dfrac { 1 } { 2 } , - 36 \right )$
Standard form
$y = 4 \left ( x + \dfrac { 1 } { 2 } \right ) ^ { 2 } - 36$
$\begin{array} {l} x = \dfrac { 5 } { 2 } \\ x = - \dfrac { 7 } { 2 } \end{array}$
Find solution by method of factorization
$\left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) \color{#FF6800}{ - } \color{#FF6800}{ 32 } = 0$
$ $ Expand the expression $ $
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 35 } = 0$
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 35 } = 0$
$acx^{2} + \left(ad + bc\right)x +bd = \left(ax + b\right)\left(cx+d\right)$
$\left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 7 } \right ) = 0$
$\left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \right ) \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 7 } \right ) = \color{#FF6800}{ 0 }$
$ $ If the product of the factor is 0, at least one factor should be 0 $ $
$\begin{array} {l} \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 7 } = \color{#FF6800}{ 0 } \end{array}$
$\begin{array} {l} \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 7 } = \color{#FF6800}{ 0 } \end{array}$
$ $ Solve the equation to find $ x$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 5 } } { \color{#FF6800}{ 2 } } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 7 } } { \color{#FF6800}{ 2 } } } \end{array}$
$\begin{array} {l} x = \dfrac { 5 } { 2 } \\ x = - \dfrac { 7 } { 2 } \end{array}$
Calculate using the quadratic formula
$\left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) \color{#FF6800}{ - } \color{#FF6800}{ 32 } = 0$
$ $ Organize the expression $ $
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 35 } = 0$
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 35 } = \color{#FF6800}{ 0 }$
$ $ Solve the quadratic equation $ ax^{2}+bx+c=0 $ using the quadratic formula $ \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 4 } \pm \sqrt{ \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 35 } \right ) } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 4 } \pm \sqrt{ \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 35 } \right ) } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 4 } \pm \sqrt{ \color{#FF6800}{ 576 } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } } }$
$x = \dfrac { - 4 \pm \sqrt{ \color{#FF6800}{ 576 } } } { 2 \times 4 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { - 4 \pm \color{#FF6800}{ 24 } } { 2 \times 4 }$
$x = \dfrac { - 4 \pm 24 } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } }$
$ $ Multiply $ 2 $ and $ 4$
$x = \dfrac { - 4 \pm 24 } { \color{#FF6800}{ 8 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 4 } \pm \color{#FF6800}{ 24 } } { \color{#FF6800}{ 8 } } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ + } \color{#FF6800}{ 24 } } { \color{#FF6800}{ 8 } } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 24 } } { \color{#FF6800}{ 8 } } } \end{array}$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ + } \color{#FF6800}{ 24 } } { 8 } \\ x = \dfrac { - 4 - 24 } { 8 } \end{array}$
$ $ Add $ - 4 $ and $ 24$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 20 } } { 8 } \\ x = \dfrac { - 4 - 24 } { 8 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 20 } } { \color{#FF6800}{ 8 } } } \\ x = \dfrac { - 4 - 24 } { 8 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 5 } } { \color{#FF6800}{ 2 } } } \\ x = \dfrac { - 4 - 24 } { 8 } \end{array}$
$\begin{array} {l} x = \dfrac { 5 } { 2 } \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ - } \color{#FF6800}{ 24 } } { 8 } \end{array}$
$ $ Find the sum of the negative numbers $ $
$\begin{array} {l} x = \dfrac { 5 } { 2 } \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 28 } } { 8 } \end{array}$
$\begin{array} {l} x = \dfrac { 5 } { 2 } \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 28 } } { \color{#FF6800}{ 8 } } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \dfrac { 5 } { 2 } \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 7 } } { \color{#FF6800}{ 2 } } } \end{array}$
$\begin{array} {l} x = \dfrac { 5 } { 2 } \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 7 } } { 2 } \end{array}$
$ $ Move the minus sign to the front of the fraction $ $
$\begin{array} {l} x = \dfrac { 5 } { 2 } \\ x = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 7 } } { \color{#FF6800}{ 2 } } } \end{array}$
$ $ 2 real roots $ $
Find the number of solutions
$\left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) \color{#FF6800}{ - } \color{#FF6800}{ 32 } = 0$
$ $ Organize the expression $ $
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 35 } = 0$
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 35 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 35 } \right )$
$D = \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } - 4 \times 4 \times \left ( - 35 \right )$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 16 } - 4 \times 4 \times \left ( - 35 \right )$
$D = 16 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 35 } \right )$
$ $ Multiply the numbers $ $
$D = 16 + \color{#FF6800}{ 560 }$
$D = \color{#FF6800}{ 16 } \color{#FF6800}{ + } \color{#FF6800}{ 560 }$
$ $ Add $ 16 $ and $ 560$
$D = \color{#FF6800}{ 576 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 576 }$
$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $
$ $ 2 real roots $ $
$\alpha + \beta = - 1 , \alpha \beta = - \dfrac { 35 } { 4 }$
Find the sum and product of the two roots of the quadratic equation
$\left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) \color{#FF6800}{ - } \color{#FF6800}{ 32 } = 0$
$ $ Organize the expression $ $
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 35 } = 0$
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 35 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 4 } } { \color{#FF6800}{ 4 } } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 35 } } { \color{#FF6800}{ 4 } } }$
$\alpha + \beta = - \color{#FF6800}{ \dfrac { \color{#FF6800}{ 4 } } { \color{#FF6800}{ 4 } } } , \alpha \beta = \dfrac { - 35 } { 4 }$
$ $ Reduce the fraction $ $
$\alpha + \beta = - \color{#FF6800}{ 1 } , \alpha \beta = \dfrac { - 35 } { 4 }$
$\alpha + \beta = - 1 , \alpha \beta = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 35 } } { 4 }$
$ $ Move the minus sign to the front of the fraction $ $
$\alpha + \beta = - 1 , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 35 } } { \color{#FF6800}{ 4 } } }$
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