Symbol

# Calculator search results

Formula
Number of solution
Relationship between roots and coefficients
Graph
$y = 4 x - \dfrac { x ^ { 2 } + 1 } { 4 }$
$y = 3 \left ( x - 1 \right )$
$x$Intercept
$\left ( 3 \sqrt{ 7 } + 8 , 0 \right )$, $\left ( 8 - 3 \sqrt{ 7 } , 0 \right )$
$y$Intercept
$\left ( 0 , - \dfrac { 1 } { 4 } \right )$
$x$Intercept
$\left ( 1 , 0 \right )$
$y$Intercept
$\left ( 0 , - 3 \right )$
$4x- \dfrac{ x ^{ 2 } +1 }{ 4 } = 3 \left( x-1 \right)$
$\begin{array} {l} x = 2 + \sqrt{ 15 } \\ x = 2 - \sqrt{ 15 } \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } + 1 } { 4 } } = 3 \left ( x - 1 \right )$
 Calculate the expression as a fraction format 
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } } = 3 \left ( x - 1 \right )$
$- \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } = \color{#FF6800}{ 3 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right )$
 Organize the expression 
$- \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } = \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } } = \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 12 }$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = 12 x - 12$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = 12 x - 12$
$- x ^ { 2 } + 16 x - 1 = \color{#FF6800}{ 12 } \color{#FF6800}{ x } - 12$
 Move the expression to the left side and change the symbol 
$- x ^ { 2 } + 16 x - 1 \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = \color{#FF6800}{ 0 }$
 Organize the expression 
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 Change the symbols of both sides of the equation 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 Convert the quadratic expression on the left side to a perfect square format 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 11 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 11 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
 Organize the expression 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 15 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 15 }$
 Solve quadratic equations using the square root 
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } = \pm \sqrt{ \color{#FF6800}{ 15 } }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } = \pm \sqrt{ \color{#FF6800}{ 15 } }$
 Solve a solution to $x$
$\color{#FF6800}{ x } = \pm \sqrt{ \color{#FF6800}{ 15 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 }$
$\color{#FF6800}{ x } = \pm \sqrt{ \color{#FF6800}{ 15 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 15 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 15 } } \end{array}$
$\begin{array} {l} x = 2 + \sqrt{ 15 } \\ x = 2 - \sqrt{ 15 } \end{array}$
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } + 1 } { 4 } } = 3 \left ( x - 1 \right )$
 Calculate the expression as a fraction format 
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } } = 3 \left ( x - 1 \right )$
$- \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } = \color{#FF6800}{ 3 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right )$
 Organize the expression 
$- \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } = \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } } = \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 12 }$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = 12 x - 12$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = 12 x - 12$
$- x ^ { 2 } + 16 x - 1 = \color{#FF6800}{ 12 } \color{#FF6800}{ x } - 12$
 Move the expression to the left side and change the symbol 
$- x ^ { 2 } + 16 x - 1 \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = \color{#FF6800}{ 0 }$
 Organize the expression 
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 Change the symbols of both sides of the equation 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 4 \right ) \pm \sqrt{ \left ( - 4 \right ) ^ { 2 } - 4 \times 1 \times \left ( - 11 \right ) } } { 2 \times 1 }$
 Simplify Minus 
$x = \dfrac { 4 \pm \sqrt{ \left ( - 4 \right ) ^ { 2 } - 4 \times 1 \times \left ( - 11 \right ) } } { 2 \times 1 }$
$x = \dfrac { 4 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - 11 \right ) } } { 2 \times 1 }$
 Remove negative signs because negative numbers raised to even powers are positive 
$x = \dfrac { 4 \pm \sqrt{ 4 ^ { 2 } - 4 \times 1 \times \left ( - 11 \right ) } } { 2 \times 1 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 \pm \sqrt{ 4 ^ { 2 } - 4 \times 1 \times \left ( - 11 \right ) } } { 2 \times 1 } }$
 Organize the expression 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 \pm \sqrt{ 60 } } { 2 \times 1 } }$
$x = \dfrac { 4 \pm \sqrt{ \color{#FF6800}{ 60 } } } { 2 \times 1 }$
 Organize the part that can be taken out of the radical sign inside the square root symbol 
$x = \dfrac { 4 \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 15 } } } { 2 \times 1 }$
$x = \dfrac { 4 \pm 2 \sqrt{ 15 } } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
 Multiplying any number by 1 does not change the value 
$x = \dfrac { 4 \pm 2 \sqrt{ 15 } } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 \pm 2 \sqrt{ 15 } } { 2 } }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 + 2 \sqrt{ 15 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 - 2 \sqrt{ 15 } } { 2 } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 4 + 2 \sqrt{ 15 } } { 2 } } \\ x = \dfrac { 4 - 2 \sqrt{ 15 } } { 2 } \end{array}$
 Do the reduction of the fraction format 
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 2 + \sqrt{ 15 } } { 1 } } \\ x = \dfrac { 4 - 2 \sqrt{ 15 } } { 2 } \end{array}$
$\begin{array} {l} x = \dfrac { 2 + \sqrt{ 15 } } { \color{#FF6800}{ 1 } } \\ x = \dfrac { 4 - 2 \sqrt{ 15 } } { 2 } \end{array}$
 If the denominator is 1, the denominator can be removed 
$\begin{array} {l} x = \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 15 } } \\ x = \dfrac { 4 - 2 \sqrt{ 15 } } { 2 } \end{array}$
$\begin{array} {l} x = 2 + \sqrt{ 15 } \\ x = \color{#FF6800}{ \dfrac { 4 - 2 \sqrt{ 15 } } { 2 } } \end{array}$
 Do the reduction of the fraction format 
$\begin{array} {l} x = 2 + \sqrt{ 15 } \\ x = \color{#FF6800}{ \dfrac { 2 - \sqrt{ 15 } } { 1 } } \end{array}$
$\begin{array} {l} x = 2 + \sqrt{ 15 } \\ x = \dfrac { 2 - \sqrt{ 15 } } { \color{#FF6800}{ 1 } } \end{array}$
 If the denominator is 1, the denominator can be removed 
$\begin{array} {l} x = 2 + \sqrt{ 15 } \\ x = \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 15 } } \end{array}$
 2 real roots 
Find the number of solutions
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } + 1 } { 4 } } = 3 \left ( x - 1 \right )$
 Calculate the expression as a fraction format 
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } } = 3 \left ( x - 1 \right )$
$- \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } = \color{#FF6800}{ 3 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right )$
 Organize the expression 
$- \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } = \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } } = \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 12 }$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = 12 x - 12$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = 12 x - 12$
$- x ^ { 2 } + 16 x - 1 = \color{#FF6800}{ 12 } \color{#FF6800}{ x } - 12$
 Move the expression to the left side and change the symbol 
$- x ^ { 2 } + 16 x - 1 \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
$- x ^ { 2 } + \color{#FF6800}{ 16 } \color{#FF6800}{ x } - 1 \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ x } + 12 = 0$
 Calculate between similar terms 
$- x ^ { 2 } + \color{#FF6800}{ 4 } \color{#FF6800}{ x } - 1 + 12 = 0$
$- x ^ { 2 } + 4 x \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
 Add $- 1$ and $12$
$- x ^ { 2 } + 4 x + \color{#FF6800}{ 11 } = 0$
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 Change the symbols of both sides of the equation 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 Determine the number of roots using discriminant, $D=b^{2}-4ac$ from quadratic equation, $ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 11 } \right )$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - 11 \right )$
 Remove negative signs because negative numbers raised to even powers are positive 
$D = 4 ^ { 2 } - 4 \times 1 \times \left ( - 11 \right )$
$D = \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - 11 \right )$
 Calculate power 
$D = \color{#FF6800}{ 16 } - 4 \times 1 \times \left ( - 11 \right )$
$D = 16 - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times \left ( - 11 \right )$
 Multiplying any number by 1 does not change the value 
$D = 16 - 4 \times \left ( - 11 \right )$
$D = 16 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 11 } \right )$
 Multiply $- 4$ and $- 11$
$D = 16 + \color{#FF6800}{ 44 }$
$D = \color{#FF6800}{ 16 } \color{#FF6800}{ + } \color{#FF6800}{ 44 }$
 Add $16$ and $44$
$D = \color{#FF6800}{ 60 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 60 }$
 Since $D>0$ , the number of real root of the following quadratic equation is 2 
 2 real roots 
$\alpha + \beta = 4 , \alpha \beta = - 11$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } + 1 } { 4 } } = 3 \left ( x - 1 \right )$
 Calculate the expression as a fraction format 
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } } = 3 \left ( x - 1 \right )$
$- \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } = \color{#FF6800}{ 3 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right )$
 Organize the expression 
$- \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } = \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
$\color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x ^ { 2 } - 16 x + 1 } { 4 } } = \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 12 }$
$\color{#FF6800}{ - } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) = 12 x - 12$
 Change the symbol of each term in parentheses when there is a (-) symbol in front of parentheses 
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 16 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = 12 x - 12$
$- x ^ { 2 } + 16 x - 1 = \color{#FF6800}{ 12 } \color{#FF6800}{ x } - 12$
 Move the expression to the left side and change the symbol 
$- x ^ { 2 } + 16 x - 1 \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
$- x ^ { 2 } + \color{#FF6800}{ 16 } \color{#FF6800}{ x } - 1 \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ x } + 12 = 0$
 Calculate between similar terms 
$- x ^ { 2 } + \color{#FF6800}{ 4 } \color{#FF6800}{ x } - 1 + 12 = 0$
$- x ^ { 2 } + 4 x \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = 0$
 Add $- 1$ and $12$
$- x ^ { 2 } + 4 x + \color{#FF6800}{ 11 } = 0$
$\color{#FF6800}{ - } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 Change the symbols of both sides of the equation 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 11 } = \color{#FF6800}{ 0 }$
 In the quadratic equation $ax^{2}+bx+c=0$ , if the two roots are $\alpha, \beta$ , then it is $\alpha + \beta =-\dfrac{b}{a}$ , $\alpha\times\beta=\dfrac{c}{a}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 4 } { 1 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 11 } { 1 } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 4 } { 1 } } , \alpha \beta = \dfrac { - 11 } { 1 }$
 Solve the sign of a fraction with a negative sign 
$\alpha + \beta = \color{#FF6800}{ \dfrac { 4 } { 1 } } , \alpha \beta = \dfrac { - 11 } { 1 }$
$\alpha + \beta = \dfrac { 4 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { - 11 } { 1 }$
 If the denominator is 1, the denominator can be removed 
$\alpha + \beta = \color{#FF6800}{ 4 } , \alpha \beta = \dfrac { - 11 } { 1 }$
$\alpha + \beta = 4 , \alpha \beta = \dfrac { - 11 } { \color{#FF6800}{ 1 } }$
 If the denominator is 1, the denominator can be removed 
$\alpha + \beta = 4 , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ 11 }$
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