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Formula
Number of solution
Relationship between roots and coefficients
Graph
$y = 4 x ^ { 2 } - 28 x + 49$
$y = 0$
$x$Intercept
$\left ( \dfrac { 7 } { 2 } , 0 \right )$
$y$Intercept
$\left ( 0 , 49 \right )$
Minimum
$\left ( \dfrac { 7 } { 2 } , 0 \right )$
Standard form
$y = 4 \left ( x - \dfrac { 7 } { 2 } \right ) ^ { 2 }$
$4x ^{ 2 } -28x+49 = 0$
$x = \dfrac { 7 } { 2 }$
Solve quadratic equations using the square root
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 49 } = \color{#FF6800}{ 0 }$
 Divide both sides by the coefficient of the leading highest term 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 7 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 49 } { 4 } } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 7 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 49 } { 4 } } = 0$
 Express as the perfect square formula 
$\color{#FF6800}{ \dfrac { 1 } { 4 } } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
$\color{#FF6800}{ \dfrac { 1 } { 4 } } \left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
 Multiply the numbers 
$\left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
 Solve quadratic equations using the square root 
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } = \color{#FF6800}{ 0 }$
 Solve a solution to $x$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 7 } { 2 } }$
$x = \dfrac { 7 } { 2 }$
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 28 \right ) \pm \sqrt{ \left ( - 28 \right ) ^ { 2 } - 4 \times 4 \times 49 } } { 2 \times 4 }$
 Simplify Minus 
$x = \dfrac { 28 \pm \sqrt{ \left ( - 28 \right ) ^ { 2 } - 4 \times 4 \times 49 } } { 2 \times 4 }$
$x = \dfrac { 28 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 28 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 4 \times 49 } } { 2 \times 4 }$
 Remove negative signs because negative numbers raised to even powers are positive 
$x = \dfrac { 28 \pm \sqrt{ 28 ^ { 2 } - 4 \times 4 \times 49 } } { 2 \times 4 }$
$x = \dfrac { 28 \pm \sqrt{ \color{#FF6800}{ 28 } ^ { \color{#FF6800}{ 2 } } - 4 \times 4 \times 49 } } { 2 \times 4 }$
 Calculate power 
$x = \dfrac { 28 \pm \sqrt{ \color{#FF6800}{ 784 } - 4 \times 4 \times 49 } } { 2 \times 4 }$
$x = \dfrac { 28 \pm \sqrt{ 784 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 49 } } } { 2 \times 4 }$
 Multiply the numbers 
$x = \dfrac { 28 \pm \sqrt{ 784 \color{#FF6800}{ - } \color{#FF6800}{ 784 } } } { 2 \times 4 }$
$x = \dfrac { 28 \pm \sqrt{ \color{#FF6800}{ 784 } \color{#FF6800}{ - } \color{#FF6800}{ 784 } } } { 2 \times 4 }$
 Remove the two numbers if the values are the same and the signs are different 
$x = \dfrac { 28 \pm \sqrt{ 0 } } { 2 \times 4 }$
$x = \dfrac { 28 \pm \sqrt{ \color{#FF6800}{ 0 } } } { 2 \times 4 }$
$n square root$ of 0 is 0 
$x = \dfrac { 28 \pm \color{#FF6800}{ 0 } } { 2 \times 4 }$
$x = \dfrac { 28 \pm 0 } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } }$
 Multiply $2$ and $4$
$x = \dfrac { 28 \pm 0 } { \color{#FF6800}{ 8 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 28 \pm 0 } { 8 } }$
 The value will not be changed even if adding or subtracting 0 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 28 } { 8 } }$
$x = \color{#FF6800}{ \dfrac { 28 } { 8 } }$
 Do the reduction of the fraction format 
$x = \color{#FF6800}{ \dfrac { 7 } { 2 } }$
 1 real root (multiple root) 
Find the number of solutions
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 49 } = \color{#FF6800}{ 0 }$
 Determine the number of roots using discriminant, $D=b^{2}-4ac$ from quadratic equation, $ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 28 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 49 }$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 28 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 4 \times 49$
 Remove negative signs because negative numbers raised to even powers are positive 
$D = 28 ^ { 2 } - 4 \times 4 \times 49$
$D = \color{#FF6800}{ 28 } ^ { \color{#FF6800}{ 2 } } - 4 \times 4 \times 49$
 Calculate power 
$D = \color{#FF6800}{ 784 } - 4 \times 4 \times 49$
$D = 784 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 49 }$
 Multiply the numbers 
$D = 784 \color{#FF6800}{ - } \color{#FF6800}{ 784 }$
$D = \color{#FF6800}{ 784 } \color{#FF6800}{ - } \color{#FF6800}{ 784 }$
 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) 
$\alpha + \beta = 7 , \alpha \beta = \dfrac { 49 } { 4 }$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 49 } = \color{#FF6800}{ 0 }$
 In the quadratic equation $ax^{2}+bx+c=0$ , if the two roots are $\alpha, \beta$ , then it is $\alpha + \beta =-\dfrac{b}{a}$ , $\alpha\times\beta=\dfrac{c}{a}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 28 } { 4 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 49 } { 4 } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 28 } { 4 } } , \alpha \beta = \dfrac { 49 } { 4 }$
 Solve the sign of a fraction with a negative sign 
$\alpha + \beta = \color{#FF6800}{ \dfrac { 28 } { 4 } } , \alpha \beta = \dfrac { 49 } { 4 }$
$\alpha + \beta = \color{#FF6800}{ \dfrac { 28 } { 4 } } , \alpha \beta = \dfrac { 49 } { 4 }$
 Reduce the fraction 
$\alpha + \beta = \color{#FF6800}{ 7 } , \alpha \beta = \dfrac { 49 } { 4 }$
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