Symbol

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Formula
Number of solution
Relationship between roots and coefficients
Graph
$y = 2 \left ( x - 2 \right ) ^ { 2 } + 16 \left ( x - 2 \right ) + 5$
$y = 0$
$x$Intercept
$\left ( - \dfrac { 3 \sqrt{ 6 } } { 2 } - 2 , 0 \right )$, $\left ( - 2 + \dfrac { 3 \sqrt{ 6 } } { 2 } , 0 \right )$
$y$Intercept
$\left ( 0 , - 19 \right )$
Minimum
$\left ( - 2 , - 27 \right )$
Standard form
$y = 2 \left ( x + 2 \right ) ^ { 2 } - 27$
$2 \left( x-2 \right) ^{ 2 } +16 \left( x-2 \right) +5 = 0$
$\begin{array} {l} x = \dfrac { - 4 + 3 \sqrt{ 6 } } { 2 } \\ x = \dfrac { - 4 - 3 \sqrt{ 6 } } { 2 } \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) \color{#FF6800}{ + } \color{#FF6800}{ 5 } = 0$
 Organize the expression 
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 19 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 19 } = \color{#FF6800}{ 0 }$
 Divide both sides by the coefficient of the leading highest term 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 19 } { 2 } } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 19 } { 2 } } = \color{#FF6800}{ 0 }$
 Convert the quadratic expression on the left side to a perfect square format 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 19 } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 19 } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
 Organize the expression 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 27 } { 2 } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 27 } { 2 } }$
 Solve quadratic equations using the square root 
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 27 } { 2 } } }$
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 2 } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 27 } { 2 } } }$
 Solve a solution to $x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { 3 \sqrt{ 6 } } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { 3 \sqrt{ 6 } } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 \sqrt{ 6 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 \sqrt{ 6 } } { 2 } } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 \sqrt{ 6 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 \sqrt{ 6 } } { 2 } } \end{array}$
 Organize the expression 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 4 + 3 \sqrt{ 6 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 4 - 3 \sqrt{ 6 } } { 2 } } \end{array}$
$\begin{array} {l} x = \dfrac { - 4 + 3 \sqrt{ 6 } } { 2 } \\ x = \dfrac { - 4 - 3 \sqrt{ 6 } } { 2 } \end{array}$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) \color{#FF6800}{ + } \color{#FF6800}{ 5 } = 0$
 Organize the expression 
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 19 } = 0$
$x = \dfrac { - 8 \pm \sqrt{ \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times \left ( - 19 \right ) } } { 2 \times 2 }$
 Calculate power 
$x = \dfrac { - 8 \pm \sqrt{ \color{#FF6800}{ 64 } - 4 \times 2 \times \left ( - 19 \right ) } } { 2 \times 2 }$
$x = \dfrac { - 8 \pm \sqrt{ 64 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 19 } \right ) } } { 2 \times 2 }$
 Multiply the numbers 
$x = \dfrac { - 8 \pm \sqrt{ 64 + \color{#FF6800}{ 152 } } } { 2 \times 2 }$
$x = \dfrac { - 8 \pm \sqrt{ \color{#FF6800}{ 64 } \color{#FF6800}{ + } \color{#FF6800}{ 152 } } } { 2 \times 2 }$
 Add $64$ and $152$
$x = \dfrac { - 8 \pm \sqrt{ \color{#FF6800}{ 216 } } } { 2 \times 2 }$
$x = \dfrac { - 8 \pm \sqrt{ \color{#FF6800}{ 216 } } } { 2 \times 2 }$
 Organize the part that can be taken out of the radical sign inside the square root symbol 
$x = \dfrac { - 8 \pm \color{#FF6800}{ 6 } \sqrt{ \color{#FF6800}{ 6 } } } { 2 \times 2 }$
$x = \dfrac { - 8 \pm 6 \sqrt{ 6 } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } }$
 Multiply $2$ and $2$
$x = \dfrac { - 8 \pm 6 \sqrt{ 6 } } { \color{#FF6800}{ 4 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 8 \pm 6 \sqrt{ 6 } } { 4 } }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 8 + 6 \sqrt{ 6 } } { 4 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 8 - 6 \sqrt{ 6 } } { 4 } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { - 8 + 6 \sqrt{ 6 } } { 4 } } \\ x = \dfrac { - 8 - 6 \sqrt{ 6 } } { 4 } \end{array}$
 Do the reduction of the fraction format 
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { - 4 + 3 \sqrt{ 6 } } { 2 } } \\ x = \dfrac { - 8 - 6 \sqrt{ 6 } } { 4 } \end{array}$
$\begin{array} {l} x = \dfrac { - 4 + 3 \sqrt{ 6 } } { 2 } \\ x = \color{#FF6800}{ \dfrac { - 8 - 6 \sqrt{ 6 } } { 4 } } \end{array}$
 Do the reduction of the fraction format 
$\begin{array} {l} x = \dfrac { - 4 + 3 \sqrt{ 6 } } { 2 } \\ x = \color{#FF6800}{ \dfrac { - 4 - 3 \sqrt{ 6 } } { 2 } } \end{array}$
 2 real roots 
Find the number of solutions
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) \color{#FF6800}{ + } \color{#FF6800}{ 5 } = 0$
 Organize the expression 
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 19 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 19 } = \color{#FF6800}{ 0 }$
 Determine the number of roots using discriminant, $D=b^{2}-4ac$ from quadratic equation, $ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 19 } \right )$
$D = \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times \left ( - 19 \right )$
 Calculate power 
$D = \color{#FF6800}{ 64 } - 4 \times 2 \times \left ( - 19 \right )$
$D = 64 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 19 } \right )$
 Multiply the numbers 
$D = 64 + \color{#FF6800}{ 152 }$
$D = \color{#FF6800}{ 64 } \color{#FF6800}{ + } \color{#FF6800}{ 152 }$
 Add $64$ and $152$
$D = \color{#FF6800}{ 216 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 216 }$
 Since $D>0$ , the number of real root of the following quadratic equation is 2 
 2 real roots 
$\alpha + \beta = - 4 , \alpha \beta = - \dfrac { 19 } { 2 }$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) \color{#FF6800}{ + } \color{#FF6800}{ 5 } = 0$
 Organize the expression 
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 19 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 19 } = \color{#FF6800}{ 0 }$
 In the quadratic equation $ax^{2}+bx+c=0$ , if the two roots are $\alpha, \beta$ , then it is $\alpha + \beta =-\dfrac{b}{a}$ , $\alpha\times\beta=\dfrac{c}{a}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 8 } { 2 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 19 } { 2 } }$
$\alpha + \beta = - \color{#FF6800}{ \dfrac { 8 } { 2 } } , \alpha \beta = \dfrac { - 19 } { 2 }$
 Reduce the fraction 
$\alpha + \beta = - \color{#FF6800}{ 4 } , \alpha \beta = \dfrac { - 19 } { 2 }$
$\alpha + \beta = - 4 , \alpha \beta = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 19 } } { 2 }$
 Move the minus sign to the front of the fraction 
$\alpha + \beta = - 4 , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 19 } { 2 } }$
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