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Formula

Solve the quadratic equation

Answer

Number of solution

Answer

Relationship between roots and coefficients

Answer

Graph

$y = 4 x ^ { 2 } - 24 x + 36$

$y = 0$

$x$Intercept

$\left ( 3 , 0 \right )$

$y$Intercept

$\left ( 0 , 36 \right )$

Minimum

$\left ( 3 , 0 \right )$

Standard form

$y = 4 \left ( x - 3 \right ) ^ { 2 }$

$4x ^{ 2 } -24x+36 = 0$

$x = 3$

Solve quadratic equations using the square root

$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 24 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 36 } = \color{#FF6800}{ 0 }$

$ $ Divide both sides by the coefficient of the leading highest term $ $

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } = \color{#FF6800}{ 0 }$

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } = 0$

$ $ Express as the perfect square formula $ $

$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = 0$

$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$

$ $ Solve quadratic equations using the square root $ $

$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 }$

$ $ Solve a solution to $ x$

$\color{#FF6800}{ x } = \color{#FF6800}{ 3 }$

$x = 3$

Calculate using the quadratic formula

$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 24 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 36 } = 0$

$ $ Bind the expressions with the common factor $ 4$

$\color{#FF6800}{ 4 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } \right ) = 0$

$\color{#FF6800}{ 4 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } \right ) = \color{#FF6800}{ 0 }$

$ $ Divide both sides by $ 4$

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } = \color{#FF6800}{ 0 }$

$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 6 \right ) \pm \sqrt{ \left ( - 6 \right ) ^ { 2 } - 4 \times 1 \times 9 } } { 2 \times 1 }$

$ $ Simplify Minus $ $

$x = \dfrac { 6 \pm \sqrt{ \left ( - 6 \right ) ^ { 2 } - 4 \times 1 \times 9 } } { 2 \times 1 }$

$x = \dfrac { 6 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 6 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 9 } } { 2 \times 1 }$

$ $ Remove negative signs because negative numbers raised to even powers are positive $ $

$x = \dfrac { 6 \pm \sqrt{ 6 ^ { 2 } - 4 \times 1 \times 9 } } { 2 \times 1 }$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm \sqrt{ 6 ^ { 2 } - 4 \times 1 \times 9 } } { 2 \times 1 } }$

$ $ Organize the expression $ $

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm \sqrt{ 0 } } { 2 \times 1 } }$

$x = \dfrac { 6 \pm \sqrt{ \color{#FF6800}{ 0 } } } { 2 \times 1 }$

$n square root $ of 0 is 0 $ $

$x = \dfrac { 6 \pm \color{#FF6800}{ 0 } } { 2 \times 1 }$

$x = \dfrac { 6 \pm 0 } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$

$ $ Multiplying any number by 1 does not change the value $ $

$x = \dfrac { 6 \pm 0 } { \color{#FF6800}{ 2 } }$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm 0 } { 2 } }$

$ $ The value will not be changed even if adding or subtracting 0 $ $

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 } { 2 } }$

$x = \color{#FF6800}{ \dfrac { 6 } { 2 } }$

$ $ Do the reduction of the fraction format $ $

$x = \color{#FF6800}{ \dfrac { 3 } { 1 } }$

$ $ Reduce the fraction to the lowest term $ $

$x = \color{#FF6800}{ 3 }$

$ $ 1 real root (multiple root) $ $

Find the number of solutions

$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$

$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 24 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 36 }$

$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 24 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 4 \times 36$

$ $ Remove negative signs because negative numbers raised to even powers are positive $ $

$D = 24 ^ { 2 } - 4 \times 4 \times 36$

$D = \color{#FF6800}{ 24 } ^ { \color{#FF6800}{ 2 } } - 4 \times 4 \times 36$

$ $ Calculate power $ $

$D = \color{#FF6800}{ 576 } - 4 \times 4 \times 36$

$D = 576 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 36 }$

$ $ Multiply the numbers $ $

$D = 576 \color{#FF6800}{ - } \color{#FF6800}{ 576 }$

$D = \color{#FF6800}{ 576 } \color{#FF6800}{ - } \color{#FF6800}{ 576 }$

$ $ Remove the two numbers if the values are the same and the signs are different $ $

$D = 0$

$\color{#FF6800}{ D } = \color{#FF6800}{ 0 }$

$ $ Since $ D=0 $ , the number of real root of the following quadratic equation is 1 (multiple root) $ $

$ $ 1 real root (multiple root) $ $

$\alpha + \beta = 6 , \alpha \beta = 9$

Find the sum and product of the two roots of the quadratic equation

$ $ 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}$

$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 24 } { 4 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 36 } { 4 } }$

$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 24 } { 4 } } , \alpha \beta = \dfrac { 36 } { 4 }$

$ $ Solve the sign of a fraction with a negative sign $ $

$\alpha + \beta = \color{#FF6800}{ \dfrac { 24 } { 4 } } , \alpha \beta = \dfrac { 36 } { 4 }$

$ $ Reduce the fraction $ $

$\alpha + \beta = \color{#FF6800}{ 6 } , \alpha \beta = \dfrac { 36 } { 4 }$

$\alpha + \beta = 6 , \alpha \beta = \color{#FF6800}{ \dfrac { 36 } { 4 } }$

$ $ Reduce the fraction $ $

$\alpha + \beta = 6 , \alpha \beta = \color{#FF6800}{ 9 }$

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