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Formula
Number of solution
Relationship between roots and coefficients
Graph
$y = x ^ { 2 } - 5 x - 6$
$y = 3$
$x$-intercept
$\left ( 6 , 0 \right )$, $\left ( - 1 , 0 \right )$
$y$-intercept
$\left ( 0 , - 6 \right )$
Minimum
$\left ( \dfrac { 5 } { 2 } , - \dfrac { 49 } { 4 } \right )$
Standard form
$y = \left ( x - \dfrac { 5 } { 2 } \right ) ^ { 2 } - \dfrac { 49 } { 4 }$
$x ^{ 2 } -5x-6=03$
$\begin{array} {l} x = \dfrac { 5 + \sqrt{ 61 } } { 2 } \\ x = \dfrac { 5 - \sqrt{ 61 } } { 2 } \end{array}$
Solve quadratic equations using the square root
$x ^ { 2 } - 5 x - 6 = \color{#FF6800}{ 3 }$
 Move the expression to the left side and change the symbol 
$x ^ { 2 } - 5 x - 6 \color{#FF6800}{ - } \color{#FF6800}{ 3 } = 0$
$x ^ { 2 } - 5 x \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = 0$
 Find the sum of the negative numbers 
$x ^ { 2 } - 5 x \color{#FF6800}{ - } \color{#FF6800}{ 9 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 9 } = \color{#FF6800}{ 0 }$
 Convert the quadratic expression on the left side to a perfect square format 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 5 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( x - \dfrac { 5 } { 2 } \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 5 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
 Move the constant to the right side and change the sign 
$\left ( x - \dfrac { 5 } { 2 } \right ) ^ { 2 } = \color{#FF6800}{ 9 } \color{#FF6800}{ + } \left ( \color{#FF6800}{ \dfrac { 5 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } }$
$\left ( x - \dfrac { 5 } { 2 } \right ) ^ { 2 } = 9 + \left ( \color{#FF6800}{ \dfrac { 5 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } }$
 When raising a fraction to the power, raise the numerator and denominator each to the power 
$\left ( x - \dfrac { 5 } { 2 } \right ) ^ { 2 } = 9 + \dfrac { \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 9 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 5 ^ { 2 } } { 2 ^ { 2 } } }$
 Organize the expression 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 61 } { 4 } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 61 } { 4 } }$
 Solve quadratic equations using the square root 
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 2 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 61 } { 4 } } }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 5 } { 2 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 61 } { 4 } } }$
 Solve a solution to $x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { \sqrt{ 61 } } { 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 5 } { 2 } }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { \sqrt{ 61 } } { 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 5 } { 2 } }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 5 } { 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \sqrt{ 61 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 5 } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \sqrt{ 61 } } { 2 } } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 5 } { 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \sqrt{ 61 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 5 } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \sqrt{ 61 } } { 2 } } \end{array}$
 Organize the expression 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 5 + \sqrt{ 61 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 5 - \sqrt{ 61 } } { 2 } } \end{array}$
$\begin{array} {l} x = \dfrac { 5 + \sqrt{ 61 } } { 2 } \\ x = \dfrac { 5 - \sqrt{ 61 } } { 2 } \end{array}$
$x ^ { 2 } - 5 x - 6 = \color{#FF6800}{ 3 }$
 Move the expression to the left side and change the symbol 
$x ^ { 2 } - 5 x - 6 \color{#FF6800}{ - } \color{#FF6800}{ 3 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 }$
 Organize the expression 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 5 \pm \sqrt{ \left ( - 5 \right ) ^ { 2 } - 4 \times 1 \times \left ( - 9 \right ) } } { 2 \times 1 } }$
$x = \dfrac { 5 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 5 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - 9 \right ) } } { 2 \times 1 }$
 Remove negative signs because negative numbers raised to even powers are positive 
$x = \dfrac { 5 \pm \sqrt{ 5 ^ { 2 } - 4 \times 1 \times \left ( - 9 \right ) } } { 2 \times 1 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 5 \pm \sqrt{ 5 ^ { 2 } - 4 \times 1 \times \left ( - 9 \right ) } } { 2 \times 1 } }$
 Organize the expression 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 5 \pm \sqrt{ 61 } } { 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 5 \pm \sqrt{ 61 } } { 2 } }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 5 + \sqrt{ 61 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 5 - \sqrt{ 61 } } { 2 } } \end{array}$
 2 real roots 
Find the number of solutions
$x ^ { 2 } - 5 x - 6 = \color{#FF6800}{ 3 }$
 Move the expression to the left side and change the symbol 
$x ^ { 2 } - 5 x - 6 \color{#FF6800}{ - } \color{#FF6800}{ 3 } = 0$
$x ^ { 2 } - 5 x \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = 0$
 Find the sum of the negative numbers 
$x ^ { 2 } - 5 x \color{#FF6800}{ - } \color{#FF6800}{ 9 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 9 } = \color{#FF6800}{ 0 }$
 Determine the number of roots using discriminant, $D=b^{2}-4ac$ from quadratic equation, $ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 5 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 9 } \right )$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 5 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - 9 \right )$
 Remove negative signs because negative numbers raised to even powers are positive 
$D = 5 ^ { 2 } - 4 \times 1 \times \left ( - 9 \right )$
$D = \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - 9 \right )$
 Calculate power 
$D = \color{#FF6800}{ 25 } - 4 \times 1 \times \left ( - 9 \right )$
$D = 25 - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times \left ( - 9 \right )$
 Multiplying any number by 1 does not change the value 
$D = 25 - 4 \times \left ( - 9 \right )$
$D = 25 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 9 } \right )$
 Multiply $- 4$ and $- 9$
$D = 25 + \color{#FF6800}{ 36 }$
$D = \color{#FF6800}{ 25 } \color{#FF6800}{ + } \color{#FF6800}{ 36 }$
 Add $25$ and $36$
$D = \color{#FF6800}{ 61 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 61 }$
 Since $D>0$ , the number of real root of the following quadratic equation is 2 
 2 real roots 
$\alpha + \beta = 5 , \alpha \beta = - 9$
Find the sum and product of the two roots of the quadratic equation
$x ^ { 2 } - 5 x - 6 = \color{#FF6800}{ 3 }$
 Move the expression to the left side and change the symbol 
$x ^ { 2 } - 5 x - 6 \color{#FF6800}{ - } \color{#FF6800}{ 3 } = 0$
$x ^ { 2 } - 5 x \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = 0$
 Find the sum of the negative numbers 
$x ^ { 2 } - 5 x \color{#FF6800}{ - } \color{#FF6800}{ 9 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 9 } = \color{#FF6800}{ 0 }$
 In the quadratic equation $ax^{2}+bx+c=0$ , if the two roots are $\alpha, \beta$ , then it is $\alpha + \beta =-\dfrac{b}{a}$ , $\alpha\times\beta=\dfrac{c}{a}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 5 } { 1 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 9 } { 1 } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 5 } { 1 } } , \alpha \beta = \dfrac { - 9 } { 1 }$
 Solve the sign of a fraction with a negative sign 
$\alpha + \beta = \color{#FF6800}{ \dfrac { 5 } { 1 } } , \alpha \beta = \dfrac { - 9 } { 1 }$
$\alpha + \beta = \dfrac { 5 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { - 9 } { 1 }$
 If the denominator is 1, the denominator can be removed 
$\alpha + \beta = \color{#FF6800}{ 5 } , \alpha \beta = \dfrac { - 9 } { 1 }$
$\alpha + \beta = 5 , \alpha \beta = \dfrac { - 9 } { \color{#FF6800}{ 1 } }$
 If the denominator is 1, the denominator can be removed 
$\alpha + \beta = 5 , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ 9 }$
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