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Formula
Number of solution
Relationship between roots and coefficients
Graph
$y = 9 x ^ { 2 } + 12 x + 4$
$y = 0$
$x$Intercept
$\left ( - \dfrac { 2 } { 3 } , 0 \right )$
$y$Intercept
$\left ( 0 , 4 \right )$
Minimum
$\left ( - \dfrac { 2 } { 3 } , 0 \right )$
Standard form
$y = 9 \left ( x + \dfrac { 2 } { 3 } \right ) ^ { 2 }$
$9x ^{ 2 } +12x+4 = 0$
$x = - \dfrac { 2 } { 3 }$
Solve quadratic equations using the square root
$\color{#FF6800}{ 9 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 4 } = \color{#FF6800}{ 0 }$
 Divide both sides by the coefficient of the leading highest term 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 4 } { 3 } } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 4 } { 9 } } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 4 } { 3 } } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 4 } { 9 } } = 0$
 Express as the perfect square formula 
$\color{#FF6800}{ \dfrac { 1 } { 9 } } \left ( \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
$\color{#FF6800}{ \dfrac { 1 } { 9 } } \left ( \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
 Multiply the numbers 
$\left ( \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
 Solve quadratic equations using the square root 
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } = \color{#FF6800}{ 0 }$
 Solve a solution to $x$
$\color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } }$
$x = - \dfrac { 2 } { 3 }$
$x = \dfrac { - 12 \pm \sqrt{ \color{#FF6800}{ 12 } ^ { \color{#FF6800}{ 2 } } - 4 \times 9 \times 4 } } { 2 \times 9 }$
 Calculate power 
$x = \dfrac { - 12 \pm \sqrt{ \color{#FF6800}{ 144 } - 4 \times 9 \times 4 } } { 2 \times 9 }$
$x = \dfrac { - 12 \pm \sqrt{ 144 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 9 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } } } { 2 \times 9 }$
 Multiply the numbers 
$x = \dfrac { - 12 \pm \sqrt{ 144 \color{#FF6800}{ - } \color{#FF6800}{ 144 } } } { 2 \times 9 }$
$x = \dfrac { - 12 \pm \sqrt{ \color{#FF6800}{ 144 } \color{#FF6800}{ - } \color{#FF6800}{ 144 } } } { 2 \times 9 }$
 Remove the two numbers if the values are the same and the signs are different 
$x = \dfrac { - 12 \pm \sqrt{ 0 } } { 2 \times 9 }$
$x = \dfrac { - 12 \pm \sqrt{ \color{#FF6800}{ 0 } } } { 2 \times 9 }$
$n square root$ of 0 is 0 
$x = \dfrac { - 12 \pm \color{#FF6800}{ 0 } } { 2 \times 9 }$
$x = \dfrac { - 12 \pm 0 } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 9 } }$
 Multiply $2$ and $9$
$x = \dfrac { - 12 \pm 0 } { \color{#FF6800}{ 18 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 12 \pm 0 } { 18 } }$
 The value will not be changed even if adding or subtracting 0 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 12 } { 18 } }$
$x = \color{#FF6800}{ \dfrac { - 12 } { 18 } }$
 Do the reduction of the fraction format 
$x = \color{#FF6800}{ \dfrac { - 2 } { 3 } }$
$x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 2 } } { 3 }$
 Move the minus sign to the front of the fraction 
$x = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 3 } }$
 1 real root (multiple root) 
Find the number of solutions
$\color{#FF6800}{ 9 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 4 } = \color{#FF6800}{ 0 }$
 Determine the number of roots using discriminant, $D=b^{2}-4ac$ from quadratic equation, $ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 12 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 9 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 }$
$D = \color{#FF6800}{ 12 } ^ { \color{#FF6800}{ 2 } } - 4 \times 9 \times 4$
 Calculate power 
$D = \color{#FF6800}{ 144 } - 4 \times 9 \times 4$
$D = 144 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 9 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 }$
 Multiply the numbers 
$D = 144 \color{#FF6800}{ - } \color{#FF6800}{ 144 }$
$D = \color{#FF6800}{ 144 } \color{#FF6800}{ - } \color{#FF6800}{ 144 }$
 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) 
$\alpha + \beta = - \dfrac { 4 } { 3 } , \alpha \beta = \dfrac { 4 } { 9 }$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 9 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 4 } = \color{#FF6800}{ 0 }$
 In the quadratic equation $ax^{2}+bx+c=0$ , if the two roots are $\alpha, \beta$ , then it is $\alpha + \beta =-\dfrac{b}{a}$ , $\alpha\times\beta=\dfrac{c}{a}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 12 } { 9 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 4 } { 9 } }$
$\alpha + \beta = - \color{#FF6800}{ \dfrac { 12 } { 9 } } , \alpha \beta = \dfrac { 4 } { 9 }$
 Reduce the fraction 
$\alpha + \beta = - \color{#FF6800}{ \dfrac { 4 } { 3 } } , \alpha \beta = \dfrac { 4 } { 9 }$
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