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Formula
Number of solution
Relationship between roots and coefficients
Graph
$y = \dfrac { x ^ { 2 } - 2 } { 3 } - \dfrac { x ^ { 2 } - 1 } { 2 }$
$y = - 2$
$y$Intercept
$\left ( 0 , - \dfrac { 1 } { 6 } \right )$
$\dfrac{ x ^{ 2 } -2 }{ 3 } - \dfrac{ x ^{ 2 } -1 }{ 2 } = -2$
$\begin{array} {l} x = \sqrt{ 11 } \\ x = - \sqrt{ 11 } \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ \dfrac { x ^ { 2 } - 2 } { 3 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } - 1 } { 2 } } = - 2$
 Calculate the expression as a fraction format 
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } + 1 } { 6 } } = - 2$
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } + 1 } { 6 } } = \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = \color{#FF6800}{ - } \color{#FF6800}{ 12 }$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = - 12$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = - 12$
$- x ^ { 2 } - 1 = \color{#FF6800}{ - } \color{#FF6800}{ 12 }$
 Move the expression to the left side and change the symbol 
$- x ^ { 2 } - 1 \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
$- x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
 Add $- 1$ and $12$
$- x ^ { 2 } + \color{#FF6800}{ 11 } = 0$
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 Change the symbols of both sides of the equation 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
$x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = 0$
 Move the constant to the right side and change the sign 
$x ^ { 2 } = \color{#FF6800}{ 11 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 11 }$
 Solve quadratic equations using the square root 
$\color{#FF6800}{ x } = \pm \sqrt{ \color{#FF6800}{ 11 } }$
$\color{#FF6800}{ x } = \pm \sqrt{ \color{#FF6800}{ 11 } }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \sqrt{ \color{#FF6800}{ 11 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 11 } } \end{array}$
$\begin{array} {l} x = \sqrt{ 11 } \\ x = - \sqrt{ 11 } \end{array}$
$\color{#FF6800}{ \dfrac { x ^ { 2 } - 2 } { 3 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } - 1 } { 2 } } = - 2$
 Calculate the expression as a fraction format 
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } + 1 } { 6 } } = - 2$
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } + 1 } { 6 } } = \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = \color{#FF6800}{ - } \color{#FF6800}{ 12 }$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = - 12$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = - 12$
$- x ^ { 2 } - 1 = \color{#FF6800}{ - } \color{#FF6800}{ 12 }$
 Move the expression to the left side and change the symbol 
$- x ^ { 2 } - 1 \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
$- x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
 Add $- 1$ and $12$
$- x ^ { 2 } + \color{#FF6800}{ 11 } = 0$
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 Change the symbols of both sides of the equation 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
$x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 0 } \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times \left ( - 11 \right ) } } { 2 \times 1 }$
 0 has no sign 
$x = \dfrac { \color{#FF6800}{ 0 } \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times \left ( - 11 \right ) } } { 2 \times 1 }$
$x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - 11 \right ) } } { 2 \times 1 }$
 The power of 0 is 0 
$x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } - 4 \times 1 \times \left ( - 11 \right ) } } { 2 \times 1 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 \pm \sqrt{ 0 - 4 \times 1 \times \left ( - 11 \right ) } } { 2 \times 1 } }$
 Organize the expression 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 \pm \sqrt{ 44 } } { 2 \times 1 } }$
$x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 44 } } } { 2 \times 1 }$
 Organize the part that can be taken out of the radical sign inside the square root symbol 
$x = \dfrac { 0 \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 11 } } } { 2 \times 1 }$
$x = \dfrac { 0 \pm 2 \sqrt{ 11 } } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
 Multiplying any number by 1 does not change the value 
$x = \dfrac { 0 \pm 2 \sqrt{ 11 } } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 \pm 2 \sqrt{ 11 } } { 2 } }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 + 2 \sqrt{ 11 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 0 - 2 \sqrt{ 11 } } { 2 } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 0 + 2 \sqrt{ 11 } } { 2 } } \\ x = \dfrac { 0 - 2 \sqrt{ 11 } } { 2 } \end{array}$
 Simplify the fraction 
$\begin{array} {l} x = \color{#FF6800}{ 0 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 11 } } \\ x = \dfrac { 0 - 2 \sqrt{ 11 } } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ 0 } + \sqrt{ 11 } \\ x = \dfrac { 0 - 2 \sqrt{ 11 } } { 2 } \end{array}$
 0 does not change when you add or subtract 
$\begin{array} {l} x = \sqrt{ 11 } \\ x = \dfrac { 0 - 2 \sqrt{ 11 } } { 2 } \end{array}$
$\begin{array} {l} x = \sqrt{ 11 } \\ x = \color{#FF6800}{ \dfrac { 0 - 2 \sqrt{ 11 } } { 2 } } \end{array}$
 Simplify the fraction 
$\begin{array} {l} x = \sqrt{ 11 } \\ x = \color{#FF6800}{ 0 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 11 } } \end{array}$
$\begin{array} {l} x = \sqrt{ 11 } \\ x = \color{#FF6800}{ 0 } - \sqrt{ 11 } \end{array}$
 0 does not change when you add or subtract 
$\begin{array} {l} x = \sqrt{ 11 } \\ x = - \sqrt{ 11 } \end{array}$
 2 real roots 
Find the number of solutions
$\color{#FF6800}{ \dfrac { x ^ { 2 } - 2 } { 3 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } - 1 } { 2 } } = - 2$
 Calculate the expression as a fraction format 
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } + 1 } { 6 } } = - 2$
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } + 1 } { 6 } } = \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = \color{#FF6800}{ - } \color{#FF6800}{ 12 }$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = - 12$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = - 12$
$- x ^ { 2 } - 1 = \color{#FF6800}{ - } \color{#FF6800}{ 12 }$
 Move the expression to the left side and change the symbol 
$- x ^ { 2 } - 1 \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
$- x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
 Add $- 1$ and $12$
$- x ^ { 2 } + \color{#FF6800}{ 11 } = 0$
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 Change the symbols of both sides of the equation 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 Determine the number of roots using discriminant, $D=b^{2}-4ac$ from quadratic equation, $ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 11 } \right )$
$D = \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - 11 \right )$
 The power of 0 is 0 
$D = \color{#FF6800}{ 0 } - 4 \times 1 \times \left ( - 11 \right )$
$D = \color{#FF6800}{ 0 } - 4 \times 1 \times \left ( - 11 \right )$
 0 does not change when you add or subtract 
$D = - 4 \times 1 \times \left ( - 11 \right )$
$D = - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times \left ( - 11 \right )$
 Multiplying any number by 1 does not change the value 
$D = - 4 \times \left ( - 11 \right )$
$D = \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 11 } \right )$
 Multiply $- 4$ and $- 11$
$D = \color{#FF6800}{ 44 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 44 }$
 Since $D>0$ , the number of real root of the following quadratic equation is 2 
 2 real roots 
$\alpha + \beta = 0 , \alpha \beta = - 11$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ \dfrac { x ^ { 2 } - 2 } { 3 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } - 1 } { 2 } } = - 2$
 Calculate the expression as a fraction format 
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } + 1 } { 6 } } = - 2$
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } + 1 } { 6 } } = \color{#FF6800}{ - } \color{#FF6800}{ 2 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = \color{#FF6800}{ - } \color{#FF6800}{ 12 }$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = - 12$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = - 12$
$- x ^ { 2 } - 1 = \color{#FF6800}{ - } \color{#FF6800}{ 12 }$
 Move the expression to the left side and change the symbol 
$- x ^ { 2 } - 1 \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
$- x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
 Add $- 1$ and $12$
$- x ^ { 2 } + \color{#FF6800}{ 11 } = 0$
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 Change the symbols of both sides of the equation 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 In the quadratic equation $ax^{2}+bx+c=0$ , if the two roots are $\alpha, \beta$ , then it is $\alpha + \beta =-\dfrac{b}{a}$ , $\alpha\times\beta=\dfrac{c}{a}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 0 } { 1 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 11 } { 1 } }$
$\alpha + \beta = - \dfrac { 0 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { - 11 } { 1 }$
 If the denominator is 1, the denominator can be removed 
$\alpha + \beta = - \color{#FF6800}{ 0 } , \alpha \beta = \dfrac { - 11 } { 1 }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ 0 } , \alpha \beta = \dfrac { - 11 } { 1 }$
 0 has no sign 
$\alpha + \beta = \color{#FF6800}{ 0 } , \alpha \beta = \dfrac { - 11 } { 1 }$
$\alpha + \beta = 0 , \alpha \beta = \dfrac { - 11 } { \color{#FF6800}{ 1 } }$
 If the denominator is 1, the denominator can be removed 
$\alpha + \beta = 0 , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ 11 }$
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