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Formula
Solve the equation
Number of solution
Relationship between roots and coefficients
Graph
$y = 3 \left ( x + 2 \right ) ^ { 2 }$
$y = 0$
$x$Intercept
$\left ( - 2 , 0 \right )$
$y$Intercept
$\left ( 0 , 12 \right )$
Minimum
$\left ( - 2 , 0 \right )$
Standard form
$y = 3 \left ( x + 2 \right ) ^ { 2 }$
$3 \left( x+2 \right) ^{ 2 } = 0$
$x = - 2$
Solve the equation
$\color{#FF6800}{ 3 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
 It should be $x + 2 = 0$
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } = \color{#FF6800}{ 0 } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } = \color{#FF6800}{ 0 } \end{array}$
 Solve each equation 
$\color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
 1 real root (multiple root) 
Find the number of solutions
$\color{#FF6800}{ 3 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
 Organize the expression 
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = \color{#FF6800}{ 0 }$
 Determine the number of roots using discriminant, $D=b^{2}-4ac$ from quadratic equation, $ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 12 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 12 }$
$D = \color{#FF6800}{ 12 } ^ { \color{#FF6800}{ 2 } } - 4 \times 3 \times 12$
 Calculate power 
$D = \color{#FF6800}{ 144 } - 4 \times 3 \times 12$
$D = 144 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 12 }$
 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) 
$\alpha + \beta = - 4 , \alpha \beta = 4$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 3 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
 Organize the expression 
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = \color{#FF6800}{ 0 }$
 In the quadratic equation $ax^{2}+bx+c=0$ , if the two roots are $\alpha, \beta$ , then it is $\alpha + \beta =-\dfrac{b}{a}$ , $\alpha\times\beta=\dfrac{c}{a}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 12 } { 3 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 12 } { 3 } }$
$\alpha + \beta = - \color{#FF6800}{ \dfrac { 12 } { 3 } } , \alpha \beta = \dfrac { 12 } { 3 }$
 Reduce the fraction 
$\alpha + \beta = - \color{#FF6800}{ 4 } , \alpha \beta = \dfrac { 12 } { 3 }$
$\alpha + \beta = - 4 , \alpha \beta = \color{#FF6800}{ \dfrac { 12 } { 3 } }$
 Reduce the fraction 
$\alpha + \beta = - 4 , \alpha \beta = \color{#FF6800}{ 4 }$
 그래프 보기 
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