qanda-logo
apple logogoogle play logo

Calculator search results

Formula
Solve the quadratic equation
Answer
circle-check-icon
expand-arrow-icon
expand-arrow-icon
Number of solution
Answer
circle-check-icon
Relationship between roots and coefficients
Answer
circle-check-icon
Graph
$y = x ^ { 2 } + 20 x + 12$
$y = 0$
$x$Intercept
$\left ( - 10 - 2 \sqrt{ 22 } , 0 \right )$, $\left ( - 10 + 2 \sqrt{ 22 } , 0 \right )$
$y$Intercept
$\left ( 0 , 12 \right )$
Minimum
$\left ( - 10 , - 88 \right )$
Standard form
$y = \left ( x + 10 \right ) ^ { 2 } - 88$
$x ^{ 2 } +20x+12 = 0$
$\begin{array} {l} x = - 10 + 2 \sqrt{ 22 } \\ x = - 10 - 2 \sqrt{ 22 } \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 20 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = \color{#FF6800}{ 0 }$
$ $ Convert the quadratic expression on the left side to a perfect square format $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 10 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( x + 10 \right ) ^ { 2 } \color{#FF6800}{ + } \color{#FF6800}{ 12 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } ^ { \color{#FF6800}{ 2 } } = 0$
$ $ Move the constant to the right side and change the sign $ $
$\left ( x + 10 \right ) ^ { 2 } = \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ + } \color{#FF6800}{ 10 } ^ { \color{#FF6800}{ 2 } }$
$\left ( x + 10 \right ) ^ { 2 } = - 12 + \color{#FF6800}{ 10 } ^ { \color{#FF6800}{ 2 } }$
$ $ Calculate power $ $
$\left ( x + 10 \right ) ^ { 2 } = - 12 + \color{#FF6800}{ 100 }$
$\left ( x + 10 \right ) ^ { 2 } = \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ + } \color{#FF6800}{ 100 }$
$ $ Add $ - 12 $ and $ 100$
$\left ( x + 10 \right ) ^ { 2 } = \color{#FF6800}{ 88 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 10 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 88 }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 10 } = \pm \sqrt{ \color{#FF6800}{ 88 } }$
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 10 } = \pm \sqrt{ \color{#FF6800}{ 88 } }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 22 } } \color{#FF6800}{ - } \color{#FF6800}{ 10 }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 22 } } \color{#FF6800}{ - } \color{#FF6800}{ 10 }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 22 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 22 } } \end{array}$
$\begin{array} {l} x = - 10 + 2 \sqrt{ 22 } \\ x = - 10 - 2 \sqrt{ 22 } \end{array}$
Calculate using the quadratic formula
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 20 \pm \sqrt{ 20 ^ { 2 } - 4 \times 1 \times 12 } } { 2 \times 1 } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 20 \pm \sqrt{ 352 } } { 2 \times 1 } }$
$x = \dfrac { - 20 \pm \sqrt{ \color{#FF6800}{ 352 } } } { 2 \times 1 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { - 20 \pm \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 22 } } } { 2 \times 1 }$
$x = \dfrac { - 20 \pm 4 \sqrt{ 22 } } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
$ $ Multiplying any number by 1 does not change the value $ $
$x = \dfrac { - 20 \pm 4 \sqrt{ 22 } } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 20 \pm 4 \sqrt{ 22 } } { 2 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 20 + 4 \sqrt{ 22 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 20 - 4 \sqrt{ 22 } } { 2 } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { - 20 + 4 \sqrt{ 22 } } { 2 } } \\ x = \dfrac { - 20 - 4 \sqrt{ 22 } } { 2 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { - 10 + 2 \sqrt{ 22 } } { 1 } } \\ x = \dfrac { - 20 - 4 \sqrt{ 22 } } { 2 } \end{array}$
$\begin{array} {l} x = \dfrac { - 10 + 2 \sqrt{ 22 } } { \color{#FF6800}{ 1 } } \\ x = \dfrac { - 20 - 4 \sqrt{ 22 } } { 2 } \end{array}$
$ $ If the denominator is 1, the denominator can be removed $ $
$\begin{array} {l} x = \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 22 } } \\ x = \dfrac { - 20 - 4 \sqrt{ 22 } } { 2 } \end{array}$
$\begin{array} {l} x = - 10 + 2 \sqrt{ 22 } \\ x = \color{#FF6800}{ \dfrac { - 20 - 4 \sqrt{ 22 } } { 2 } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = - 10 + 2 \sqrt{ 22 } \\ x = \color{#FF6800}{ \dfrac { - 10 - 2 \sqrt{ 22 } } { 1 } } \end{array}$
$\begin{array} {l} x = - 10 + 2 \sqrt{ 22 } \\ x = \dfrac { - 10 - 2 \sqrt{ 22 } } { \color{#FF6800}{ 1 } } \end{array}$
$ $ If the denominator is 1, the denominator can be removed $ $
$\begin{array} {l} x = - 10 + 2 \sqrt{ 22 } \\ x = \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 22 } } \end{array}$
$ $ 2 real roots $ $
Find the number of solutions
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 20 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 20 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ 12 }$
$D = \color{#FF6800}{ 20 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 12$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 400 } - 4 \times 1 \times 12$
$D = 400 - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times 12$
$ $ Multiplying any number by 1 does not change the value $ $
$D = 400 - 4 \times 12$
$D = 400 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 12 }$
$ $ Multiply $ - 4 $ and $ 12$
$D = 400 \color{#FF6800}{ - } \color{#FF6800}{ 48 }$
$D = \color{#FF6800}{ 400 } \color{#FF6800}{ - } \color{#FF6800}{ 48 }$
$ $ Subtract $ 48 $ from $ 400$
$D = \color{#FF6800}{ 352 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 352 }$
$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $
$ $ 2 real roots $ $
$\alpha + \beta = - 20 , \alpha \beta = 12$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 20 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 12 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 20 } { 1 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 12 } { 1 } }$
$\alpha + \beta = - \dfrac { 20 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { 12 } { 1 }$
$ $ If the denominator is 1, the denominator can be removed $ $
$\alpha + \beta = - \color{#FF6800}{ 20 } , \alpha \beta = \dfrac { 12 } { 1 }$
$\alpha + \beta = - 20 , \alpha \beta = \dfrac { 12 } { \color{#FF6800}{ 1 } }$
$ $ If the denominator is 1, the denominator can be removed $ $
$\alpha + \beta = - 20 , \alpha \beta = \color{#FF6800}{ 12 }$
$ $ 그래프 보기 $ $
Graph
Solution search results
search-thumbnail-Describes the relationship between the coefficients and the roots of a quadratic equation $M9A-1c-2\right)$ 
Exercise $1$ Complete the table below. 
Roots 
Equation b $x_{1}$ $x_{2}$ $x_{1}+x_{2}$ $x_{1}.x_{2}$ 
$x^{2}+6x+9=0$ $6$ 
$x^{2}-x-30=0$ $-30$ 
$x^{2}-144=0$ $1$ $0$ 

$x^{2}+20x+100=0$ 
$\dfrac {120} {2-z}$ $\dfrac {100} {12}$ 
$2x^{2}-7x+12=0$
7th-9th grade
Other
Have you found the solution you wanted?
Try again
Try more features at Qanda!
Search by problem image
Ask 1:1 question to TOP class teachers
AI recommend problems and video lecture
apple logogoogle play logo