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Solve the quadratic equation
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Number of solution
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Relationship between roots and coefficients
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Graph
$y = \left ( 4 x + 1 \right ) ^ { 2 }$
$y = 5$
$x$Intercept
$\left ( - \dfrac { 1 } { 4 } , 0 \right )$
$y$Intercept
$\left ( 0 , 1 \right )$
Minimum
$\left ( - \dfrac { 1 } { 4 } , 0 \right )$
Standard form
$y = 16 \left ( x + \dfrac { 1 } { 4 } \right ) ^ { 2 }$
$\begin{array} {l} x = \dfrac { - 1 + \sqrt{ 5 } } { 4 } \\ x = \dfrac { - 1 - \sqrt{ 5 } } { 4 } \end{array}$
Calculate using the quadratic formula
$\left ( \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = 5$
$ $ Organize the expression $ $
$\color{#FF6800}{ 16 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = 5$
$16 x ^ { 2 } + 8 x + 1 = \color{#FF6800}{ 5 }$
$ $ Move the expression to the left side and change the symbol $ $
$16 x ^ { 2 } + 8 x + 1 \color{#FF6800}{ - } \color{#FF6800}{ 5 } = 0$
$16 x ^ { 2 } + 8 x + \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = 0$
$ $ Subtract $ 5 $ from $ 1$
$16 x ^ { 2 } + 8 x \color{#FF6800}{ - } \color{#FF6800}{ 4 } = 0$
$\color{#FF6800}{ 16 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 4 } = \color{#FF6800}{ 0 }$
$ $ Solve the quadratic equation $ ax^{2}+bx+c=0 $ using the quadratic formula $ \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \pm \sqrt{ \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \pm \sqrt{ \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \pm \sqrt{ \color{#FF6800}{ 320 } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } } }$
$x = \dfrac { - 8 \pm \sqrt{ \color{#FF6800}{ 320 } } } { 2 \times 16 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { - 8 \pm \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { 2 \times 16 }$
$x = \dfrac { - 8 \pm 8 \sqrt{ 5 } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } }$
$ $ Multiply $ 2 $ and $ 16$
$x = \dfrac { - 8 \pm 8 \sqrt{ 5 } } { \color{#FF6800}{ 32 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \pm \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 32 } } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 32 } } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 32 } } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 32 } } } \\ x = \dfrac { - 8 - 8 \sqrt{ 5 } } { 32 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 4 } } } \\ x = \dfrac { - 8 - 8 \sqrt{ 5 } } { 32 } \end{array}$
$\begin{array} {l} x = \dfrac { - 1 + \sqrt{ 5 } } { 4 } \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 32 } } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \dfrac { - 1 + \sqrt{ 5 } } { 4 } \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 5 } } } { \color{#FF6800}{ 4 } } } \end{array}$
$ $ 2 real roots $ $
Find the number of solutions
$\left ( \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = 5$
$ $ Organize the expression $ $
$\color{#FF6800}{ 16 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = 5$
$16 x ^ { 2 } + 8 x + 1 = \color{#FF6800}{ 5 }$
$ $ Move the expression to the left side and change the symbol $ $
$16 x ^ { 2 } + 8 x + 1 \color{#FF6800}{ - } \color{#FF6800}{ 5 } = 0$
$16 x ^ { 2 } + 8 x + \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = 0$
$ $ Subtract $ 5 $ from $ 1$
$16 x ^ { 2 } + 8 x \color{#FF6800}{ - } \color{#FF6800}{ 4 } = 0$
$\color{#FF6800}{ 16 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 4 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right )$
$D = \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } - 4 \times 16 \times \left ( - 4 \right )$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 64 } - 4 \times 16 \times \left ( - 4 \right )$
$D = 64 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right )$
$ $ Multiply the numbers $ $
$D = 64 + \color{#FF6800}{ 256 }$
$D = \color{#FF6800}{ 64 } \color{#FF6800}{ + } \color{#FF6800}{ 256 }$
$ $ Add $ 64 $ and $ 256$
$D = \color{#FF6800}{ 320 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 320 }$
$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $
$ $ 2 real roots $ $
$\alpha + \beta = - \dfrac { 1 } { 2 } , \alpha \beta = - \dfrac { 1 } { 4 }$
Find the sum and product of the two roots of the quadratic equation
$\left ( \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = 5$
$ $ Organize the expression $ $
$\color{#FF6800}{ 16 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = 5$
$16 x ^ { 2 } + 8 x + 1 = \color{#FF6800}{ 5 }$
$ $ Move the expression to the left side and change the symbol $ $
$16 x ^ { 2 } + 8 x + 1 \color{#FF6800}{ - } \color{#FF6800}{ 5 } = 0$
$16 x ^ { 2 } + 8 x + \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = 0$
$ $ Subtract $ 5 $ from $ 1$
$16 x ^ { 2 } + 8 x \color{#FF6800}{ - } \color{#FF6800}{ 4 } = 0$
$\color{#FF6800}{ 16 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 4 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 8 } } { \color{#FF6800}{ 16 } } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 4 } } { \color{#FF6800}{ 16 } } }$
$\alpha + \beta = - \color{#FF6800}{ \dfrac { \color{#FF6800}{ 8 } } { \color{#FF6800}{ 16 } } } , \alpha \beta = \dfrac { - 4 } { 16 }$
$ $ Reduce the fraction $ $
$\alpha + \beta = - \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 2 } } } , \alpha \beta = \dfrac { - 4 } { 16 }$
$\alpha + \beta = - \dfrac { 1 } { 2 } , \alpha \beta = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 4 } } { \color{#FF6800}{ 16 } } }$
$ $ Reduce the fraction $ $
$\alpha + \beta = - \dfrac { 1 } { 2 } , \alpha \beta = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } }$
$\alpha + \beta = - \dfrac { 1 } { 2 } , \alpha \beta = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } } { 4 }$
$ $ Move the minus sign to the front of the fraction $ $
$\alpha + \beta = - \dfrac { 1 } { 2 } , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 1 } } { \color{#FF6800}{ 4 } } }$
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