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Formula
Number of solution
Relationship between roots and coefficients
Graph
$y = 2 \left ( x + 3 \right ) ^ { 2 }$
$y = 54$
$x$-intercept
$\left ( - 3 , 0 \right )$
$y$-intercept
$\left ( 0 , 18 \right )$
Minimum
$\left ( - 3 , 0 \right )$
Standard form
$y = 2 \left ( x + 3 \right ) ^ { 2 }$
$2 \left( x+3 \right) ^{ 2 } = 54$
$\begin{array} {l} x = - 3 + 3 \sqrt{ 3 } \\ x = - 3 - 3 \sqrt{ 3 } \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = 54$
 Organize the expression 
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 18 } = 54$
$2 x ^ { 2 } + 12 x + 18 = \color{#FF6800}{ 54 }$
 Move the expression to the left side and change the symbol 
$2 x ^ { 2 } + 12 x + 18 \color{#FF6800}{ - } \color{#FF6800}{ 54 } = 0$
$2 x ^ { 2 } + 12 x + \color{#FF6800}{ 18 } \color{#FF6800}{ - } \color{#FF6800}{ 54 } = 0$
 Subtract $54$ from $18$
$2 x ^ { 2 } + 12 x \color{#FF6800}{ - } \color{#FF6800}{ 36 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 36 } = \color{#FF6800}{ 0 }$
 Divide both sides by the coefficient of the leading highest term 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 18 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 18 } = \color{#FF6800}{ 0 }$
 Convert the quadratic expression on the left side to a perfect square format 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 18 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 18 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
 Organize the expression 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 27 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 27 }$
 Solve quadratic equations using the square root 
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } = \pm \sqrt{ \color{#FF6800}{ 27 } }$
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } = \pm \sqrt{ \color{#FF6800}{ 27 } }$
 Solve a solution to $x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \end{array}$
$\begin{array} {l} x = - 3 + 3 \sqrt{ 3 } \\ x = - 3 - 3 \sqrt{ 3 } \end{array}$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = 54$
 Organize the expression 
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 18 } = 54$
$2 x ^ { 2 } + 12 x + 18 = \color{#FF6800}{ 54 }$
 Move the expression to the left side and change the symbol 
$2 x ^ { 2 } + 12 x + 18 \color{#FF6800}{ - } \color{#FF6800}{ 54 } = 0$
$2 x ^ { 2 } + 12 x + \color{#FF6800}{ 18 } \color{#FF6800}{ - } \color{#FF6800}{ 54 } = 0$
 Subtract $54$ from $18$
$2 x ^ { 2 } + 12 x \color{#FF6800}{ - } \color{#FF6800}{ 36 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 36 } = 0$
 Bind the expressions with the common factor $2$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 18 } \right ) = 0$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 18 } \right ) = \color{#FF6800}{ 0 }$
 Divide both sides by $2$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 18 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 6 \pm \sqrt{ 6 ^ { 2 } - 4 \times 1 \times \left ( - 18 \right ) } } { 2 \times 1 } }$
 Organize the expression 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 6 \pm \sqrt{ 108 } } { 2 \times 1 } }$
$x = \dfrac { - 6 \pm \sqrt{ \color{#FF6800}{ 108 } } } { 2 \times 1 }$
 Organize the part that can be taken out of the radical sign inside the square root symbol 
$x = \dfrac { - 6 \pm \color{#FF6800}{ 6 } \sqrt{ \color{#FF6800}{ 3 } } } { 2 \times 1 }$
$x = \dfrac { - 6 \pm 6 \sqrt{ 3 } } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
 Multiplying any number by 1 does not change the value 
$x = \dfrac { - 6 \pm 6 \sqrt{ 3 } } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 6 \pm 6 \sqrt{ 3 } } { 2 } }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 6 + 6 \sqrt{ 3 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 6 - 6 \sqrt{ 3 } } { 2 } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { - 6 + 6 \sqrt{ 3 } } { 2 } } \\ x = \dfrac { - 6 - 6 \sqrt{ 3 } } { 2 } \end{array}$
 Do the reduction of the fraction format 
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { - 3 + 3 \sqrt{ 3 } } { 1 } } \\ x = \dfrac { - 6 - 6 \sqrt{ 3 } } { 2 } \end{array}$
$\begin{array} {l} x = \dfrac { - 3 + 3 \sqrt{ 3 } } { \color{#FF6800}{ 1 } } \\ x = \dfrac { - 6 - 6 \sqrt{ 3 } } { 2 } \end{array}$
 If the denominator is 1, the denominator can be removed 
$\begin{array} {l} x = \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \\ x = \dfrac { - 6 - 6 \sqrt{ 3 } } { 2 } \end{array}$
$\begin{array} {l} x = - 3 + 3 \sqrt{ 3 } \\ x = \color{#FF6800}{ \dfrac { - 6 - 6 \sqrt{ 3 } } { 2 } } \end{array}$
 Do the reduction of the fraction format 
$\begin{array} {l} x = - 3 + 3 \sqrt{ 3 } \\ x = \color{#FF6800}{ \dfrac { - 3 - 3 \sqrt{ 3 } } { 1 } } \end{array}$
$\begin{array} {l} x = - 3 + 3 \sqrt{ 3 } \\ x = \dfrac { - 3 - 3 \sqrt{ 3 } } { \color{#FF6800}{ 1 } } \end{array}$
 If the denominator is 1, the denominator can be removed 
$\begin{array} {l} x = - 3 + 3 \sqrt{ 3 } \\ x = \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \sqrt{ \color{#FF6800}{ 3 } } \end{array}$
 2 real roots 
Find the number of solutions
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = 54$
 Organize the expression 
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 18 } = 54$
$2 x ^ { 2 } + 12 x + 18 = \color{#FF6800}{ 54 }$
 Move the expression to the left side and change the symbol 
$2 x ^ { 2 } + 12 x + 18 \color{#FF6800}{ - } \color{#FF6800}{ 54 } = 0$
$2 x ^ { 2 } + 12 x + \color{#FF6800}{ 18 } \color{#FF6800}{ - } \color{#FF6800}{ 54 } = 0$
 Subtract $54$ from $18$
$2 x ^ { 2 } + 12 x \color{#FF6800}{ - } \color{#FF6800}{ 36 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 36 } = \color{#FF6800}{ 0 }$
 Determine the number of roots using discriminant, $D=b^{2}-4ac$ from quadratic equation, $ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 12 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 36 } \right )$
$D = \color{#FF6800}{ 12 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times \left ( - 36 \right )$
 Calculate power 
$D = \color{#FF6800}{ 144 } - 4 \times 2 \times \left ( - 36 \right )$
$D = 144 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 36 } \right )$
 Multiply the numbers 
$D = 144 + \color{#FF6800}{ 288 }$
$D = \color{#FF6800}{ 144 } \color{#FF6800}{ + } \color{#FF6800}{ 288 }$
 Add $144$ and $288$
$D = \color{#FF6800}{ 432 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 432 }$
 Since $D>0$ , the number of real root of the following quadratic equation is 2 
 2 real roots 
$\alpha + \beta = - 6 , \alpha \beta = - 18$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = 54$
 Organize the expression 
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 18 } = 54$
$2 x ^ { 2 } + 12 x + 18 = \color{#FF6800}{ 54 }$
 Move the expression to the left side and change the symbol 
$2 x ^ { 2 } + 12 x + 18 \color{#FF6800}{ - } \color{#FF6800}{ 54 } = 0$
$2 x ^ { 2 } + 12 x + \color{#FF6800}{ 18 } \color{#FF6800}{ - } \color{#FF6800}{ 54 } = 0$
 Subtract $54$ from $18$
$2 x ^ { 2 } + 12 x \color{#FF6800}{ - } \color{#FF6800}{ 36 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 36 } = \color{#FF6800}{ 0 }$
 In the quadratic equation $ax^{2}+bx+c=0$ , if the two roots are $\alpha, \beta$ , then it is $\alpha + \beta =-\dfrac{b}{a}$ , $\alpha\times\beta=\dfrac{c}{a}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 12 } { 2 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 36 } { 2 } }$
$\alpha + \beta = - \color{#FF6800}{ \dfrac { 12 } { 2 } } , \alpha \beta = \dfrac { - 36 } { 2 }$
 Reduce the fraction 
$\alpha + \beta = - \color{#FF6800}{ 6 } , \alpha \beta = \dfrac { - 36 } { 2 }$
$\alpha + \beta = - 6 , \alpha \beta = \color{#FF6800}{ \dfrac { - 36 } { 2 } }$
 Reduce the fraction 
$\alpha + \beta = - 6 , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ 18 }$
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