# Calculator search results

Formula
Number of solution
Relationship between roots and coefficients
Graph
$y = \left ( 12 - x \right ) \left ( 16 - x \right )$
$y = 117$
$x$Intercept
$\left ( 16 , 0 \right )$, $\left ( 12 , 0 \right )$
$y$Intercept
$\left ( 0 , 192 \right )$
Minimum
$\left ( 14 , - 4 \right )$
Standard form
$y = \left ( x - 14 \right ) ^ { 2 } - 4$
$\left( 12-x \right) \left( 16-x \right) = 117$
$\begin{array} {l} x = 25 \\ x = 3 \end{array}$
Find solution by method of factorization
$\left ( 12 - x \right ) \left ( 16 - x \right ) = \color{#FF6800}{ 117 }$
 Move the expression to the left side and change the symbol 
$\left ( 12 - x \right ) \left ( 16 - x \right ) - 117 = 0$
$\left ( \color{#FF6800}{ 12 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \color{#FF6800}{ - } \color{#FF6800}{ 117 } = 0$
 Expand the expression 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 75 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 75 } = 0$
$acx^{2} + \left(ad + bc\right)x +bd = \left(ax + b\right)\left(cx+d\right)$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 25 } \right ) \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) = 0$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 25 } \right ) \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) = \color{#FF6800}{ 0 }$
 If the product of the factor is 0, at least one factor should be 0 
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 25 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 25 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 } \end{array}$
 Solve the equation to find $x$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 25 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 3 } \end{array}$
$\begin{array} {l} x = 25 \\ x = 3 \end{array}$
Solve quadratic equations using the square root
$\left ( \color{#FF6800}{ 12 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) = 117$
 Organize the expression 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 192 } = 117$
$x ^ { 2 } - 28 x + 192 = \color{#FF6800}{ 117 }$
 Move the expression to the left side and change the symbol 
$x ^ { 2 } - 28 x + 192 \color{#FF6800}{ - } \color{#FF6800}{ 117 } = 0$
$x ^ { 2 } - 28 x + \color{#FF6800}{ 192 } \color{#FF6800}{ - } \color{#FF6800}{ 117 } = 0$
 Subtract $117$ from $192$
$x ^ { 2 } - 28 x + \color{#FF6800}{ 75 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 75 } = \color{#FF6800}{ 0 }$
 Convert the quadratic expression on the left side to a perfect square format 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 14 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 75 } \color{#FF6800}{ - } \color{#FF6800}{ 14 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 14 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 75 } \color{#FF6800}{ - } \color{#FF6800}{ 14 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
 Organize the expression 
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 14 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 121 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 14 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 121 }$
 Solve quadratic equations using the square root 
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 14 } = \pm \sqrt{ \color{#FF6800}{ 121 } }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 14 } = \pm \sqrt{ \color{#FF6800}{ 121 } }$
 Solve a solution to $x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 11 } \color{#FF6800}{ + } \color{#FF6800}{ 14 }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 11 } \color{#FF6800}{ + } \color{#FF6800}{ 14 }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 14 } \color{#FF6800}{ + } \color{#FF6800}{ 11 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 14 } \color{#FF6800}{ - } \color{#FF6800}{ 11 } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 14 } \color{#FF6800}{ + } \color{#FF6800}{ 11 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 14 } \color{#FF6800}{ - } \color{#FF6800}{ 11 } \end{array}$
 Organize the expression 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 25 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 3 } \end{array}$
$\begin{array} {l} x = 25 \\ x = 3 \end{array}$
$\left ( \color{#FF6800}{ 12 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) = 117$
 Organize the expression 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 192 } = 117$
$x ^ { 2 } - 28 x + 192 = \color{#FF6800}{ 117 }$
 Move the expression to the left side and change the symbol 
$x ^ { 2 } - 28 x + 192 \color{#FF6800}{ - } \color{#FF6800}{ 117 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 192 } \color{#FF6800}{ - } \color{#FF6800}{ 117 } = \color{#FF6800}{ 0 }$
 Organize the expression 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 28 \pm \sqrt{ \left ( - 28 \right ) ^ { 2 } - 4 \times 1 \times 75 } } { 2 \times 1 } }$
$x = \dfrac { 28 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 28 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 75 } } { 2 \times 1 }$
 Remove negative signs because negative numbers raised to even powers are positive 
$x = \dfrac { 28 \pm \sqrt{ 28 ^ { 2 } - 4 \times 1 \times 75 } } { 2 \times 1 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 28 \pm \sqrt{ 28 ^ { 2 } - 4 \times 1 \times 75 } } { 2 \times 1 } }$
 Organize the expression 
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 28 \pm \sqrt{ 484 } } { 2 \times 1 } }$
$x = \dfrac { 28 \pm \sqrt{ \color{#FF6800}{ 484 } } } { 2 \times 1 }$
 Organize the part that can be taken out of the radical sign inside the square root symbol 
$x = \dfrac { 28 \pm \color{#FF6800}{ 22 } } { 2 \times 1 }$
$x = \dfrac { 28 \pm 22 } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
 Multiplying any number by 1 does not change the value 
$x = \dfrac { 28 \pm 22 } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 28 \pm 22 } { 2 } }$
 Separate the answer 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 28 + 22 } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 28 - 22 } { 2 } } \end{array}$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 28 } \color{#FF6800}{ + } \color{#FF6800}{ 22 } } { 2 } \\ x = \dfrac { 28 - 22 } { 2 } \end{array}$
 Add $28$ and $22$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 50 } } { 2 } \\ x = \dfrac { 28 - 22 } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 50 } { 2 } } \\ x = \dfrac { 28 - 22 } { 2 } \end{array}$
 Do the reduction of the fraction format 
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 25 } { 1 } } \\ x = \dfrac { 28 - 22 } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 25 } { 1 } } \\ x = \dfrac { 28 - 22 } { 2 } \end{array}$
 Reduce the fraction to the lowest term 
$\begin{array} {l} x = \color{#FF6800}{ 25 } \\ x = \dfrac { 28 - 22 } { 2 } \end{array}$
$\begin{array} {l} x = 25 \\ x = \dfrac { \color{#FF6800}{ 28 } \color{#FF6800}{ - } \color{#FF6800}{ 22 } } { 2 } \end{array}$
 Subtract $22$ from $28$
$\begin{array} {l} x = 25 \\ x = \dfrac { \color{#FF6800}{ 6 } } { 2 } \end{array}$
$\begin{array} {l} x = 25 \\ x = \color{#FF6800}{ \dfrac { 6 } { 2 } } \end{array}$
 Do the reduction of the fraction format 
$\begin{array} {l} x = 25 \\ x = \color{#FF6800}{ \dfrac { 3 } { 1 } } \end{array}$
$\begin{array} {l} x = 25 \\ x = \color{#FF6800}{ \dfrac { 3 } { 1 } } \end{array}$
 Reduce the fraction to the lowest term 
$\begin{array} {l} x = 25 \\ x = \color{#FF6800}{ 3 } \end{array}$
 2 real roots 
Find the number of solutions
$\left ( \color{#FF6800}{ 12 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) = 117$
 Organize the expression 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 192 } = 117$
$x ^ { 2 } - 28 x + 192 = \color{#FF6800}{ 117 }$
 Move the expression to the left side and change the symbol 
$x ^ { 2 } - 28 x + 192 \color{#FF6800}{ - } \color{#FF6800}{ 117 } = 0$
$x ^ { 2 } - 28 x + \color{#FF6800}{ 192 } \color{#FF6800}{ - } \color{#FF6800}{ 117 } = 0$
 Subtract $117$ from $192$
$x ^ { 2 } - 28 x + \color{#FF6800}{ 75 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 75 } = \color{#FF6800}{ 0 }$
 Determine the number of roots using discriminant, $D=b^{2}-4ac$ from quadratic equation, $ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 28 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ 75 }$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 28 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 75$
 Remove negative signs because negative numbers raised to even powers are positive 
$D = 28 ^ { 2 } - 4 \times 1 \times 75$
$D = \color{#FF6800}{ 28 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 75$
 Calculate power 
$D = \color{#FF6800}{ 784 } - 4 \times 1 \times 75$
$D = 784 - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times 75$
 Multiplying any number by 1 does not change the value 
$D = 784 - 4 \times 75$
$D = 784 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 75 }$
 Multiply $- 4$ and $75$
$D = 784 \color{#FF6800}{ - } \color{#FF6800}{ 300 }$
$D = \color{#FF6800}{ 784 } \color{#FF6800}{ - } \color{#FF6800}{ 300 }$
 Subtract $300$ from $784$
$D = \color{#FF6800}{ 484 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 484 }$
 Since $D>0$ , the number of real root of the following quadratic equation is 2 
 2 real roots 
$\alpha + \beta = 28 , \alpha \beta = 75$
Find the sum and product of the two roots of the quadratic equation
$\left ( \color{#FF6800}{ 12 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) = 117$
 Organize the expression 
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 192 } = 117$
$x ^ { 2 } - 28 x + 192 = \color{#FF6800}{ 117 }$
 Move the expression to the left side and change the symbol 
$x ^ { 2 } - 28 x + 192 \color{#FF6800}{ - } \color{#FF6800}{ 117 } = 0$
$x ^ { 2 } - 28 x + \color{#FF6800}{ 192 } \color{#FF6800}{ - } \color{#FF6800}{ 117 } = 0$
 Subtract $117$ from $192$
$x ^ { 2 } - 28 x + \color{#FF6800}{ 75 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 28 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 75 } = \color{#FF6800}{ 0 }$
 In the quadratic equation $ax^{2}+bx+c=0$ , if the two roots are $\alpha, \beta$ , then it is $\alpha + \beta =-\dfrac{b}{a}$ , $\alpha\times\beta=\dfrac{c}{a}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 28 } { 1 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 75 } { 1 } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 28 } { 1 } } , \alpha \beta = \dfrac { 75 } { 1 }$
 Solve the sign of a fraction with a negative sign 
$\alpha + \beta = \color{#FF6800}{ \dfrac { 28 } { 1 } } , \alpha \beta = \dfrac { 75 } { 1 }$
$\alpha + \beta = \dfrac { 28 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { 75 } { 1 }$
 If the denominator is 1, the denominator can be removed 
$\alpha + \beta = \color{#FF6800}{ 28 } , \alpha \beta = \dfrac { 75 } { 1 }$
$\alpha + \beta = 28 , \alpha \beta = \dfrac { 75 } { \color{#FF6800}{ 1 } }$
 If the denominator is 1, the denominator can be removed 
$\alpha + \beta = 28 , \alpha \beta = \color{#FF6800}{ 75 }$
 그래프 보기 
Graph
Solution search results