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Formula

Solve the quadratic equation

Answer

Number of solution

Answer

Relationship between roots and coefficients

Answer

Graph

$y = x ^ { 2 } + 21$

$y = 10 x$

$y$Intercept

$\left ( 0 , 21 \right )$

Minimum

$\left ( 0 , 21 \right )$

Standard form

$y = x ^ { 2 } + 21$

$x$Intercept

$\left ( 0 , 0 \right )$

$y$Intercept

$\left ( 0 , 0 \right )$

$x ^{ 2 } +21 = 10x$

$\begin{array} {l} x = 7 \\ x = 3 \end{array}$

Find solution by method of factorization

$x ^ { 2 } + 21 = \color{#FF6800}{ 10 } \color{#FF6800}{ x }$

$ $ Move the expression to the left side and change the symbol $ $

$x ^ { 2 } + 21 \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } = 0$

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 21 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } = 0$

$ $ Expand the expression $ $

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 21 } = 0$

$acx^{2} + \left(ad + bc\right)x +bd = \left(ax + b\right)\left(cx+d\right)$

$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } \right ) \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) = 0$

$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } \right ) \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) = \color{#FF6800}{ 0 }$

$ $ If the product of the factor is 0, at least one factor should be 0 $ $

$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 } \end{array}$

$ $ Solve the equation to find $ x$

$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 7 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 3 } \end{array}$

$\begin{array} {l} x = 7 \\ x = 3 \end{array}$

Solve quadratic equations using the square root

$x ^ { 2 } + 21 = \color{#FF6800}{ 10 } \color{#FF6800}{ x }$

$ $ Move the expression to the left side and change the symbol $ $

$x ^ { 2 } + 21 \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } = 0$

$x ^ { 2 } \color{#FF6800}{ + } \color{#FF6800}{ 21 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } = 0$

$ $ Organize $ x $ in descending power $ $

$x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 21 } = 0$

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 21 } = \color{#FF6800}{ 0 }$

$ $ Convert the quadratic expression on the left side to a perfect square format $ $

$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 21 } \color{#FF6800}{ - } \color{#FF6800}{ 5 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$

$ $ Organize the expression $ $

$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 4 }$

$ $ Solve quadratic equations using the square root $ $

$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } = \pm \sqrt{ \color{#FF6800}{ 4 } }$

$ $ Solve a solution to $ x$

$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 5 }$

$ $ Separate the answer $ $

$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 5 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 5 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \end{array}$

$ $ Organize the expression $ $

$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 7 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 3 } \end{array}$

$\begin{array} {l} x = 7 \\ x = 3 \end{array}$

Calculate using the quadratic formula

$x ^ { 2 } + 21 = \color{#FF6800}{ 10 } \color{#FF6800}{ x }$

$ $ Move the expression to the left side and change the symbol $ $

$x ^ { 2 } + 21 \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } = 0$

$x ^ { 2 } \color{#FF6800}{ + } \color{#FF6800}{ 21 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } = 0$

$ $ Organize $ x $ in descending power $ $

$x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 21 } = 0$

$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 10 \right ) \pm \sqrt{ \left ( - 10 \right ) ^ { 2 } - 4 \times 1 \times 21 } } { 2 \times 1 }$

$ $ Simplify Minus $ $

$x = \dfrac { 10 \pm \sqrt{ \left ( - 10 \right ) ^ { 2 } - 4 \times 1 \times 21 } } { 2 \times 1 }$

$x = \dfrac { 10 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 10 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 21 } } { 2 \times 1 }$

$ $ Remove negative signs because negative numbers raised to even powers are positive $ $

$x = \dfrac { 10 \pm \sqrt{ 10 ^ { 2 } - 4 \times 1 \times 21 } } { 2 \times 1 }$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 10 \pm \sqrt{ 10 ^ { 2 } - 4 \times 1 \times 21 } } { 2 \times 1 } }$

$ $ Organize the expression $ $

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 10 \pm \sqrt{ 16 } } { 2 \times 1 } }$

$x = \dfrac { 10 \pm \sqrt{ \color{#FF6800}{ 16 } } } { 2 \times 1 }$

$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $

$x = \dfrac { 10 \pm \color{#FF6800}{ 4 } } { 2 \times 1 }$

$x = \dfrac { 10 \pm 4 } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$

$ $ Multiplying any number by 1 does not change the value $ $

$x = \dfrac { 10 \pm 4 } { \color{#FF6800}{ 2 } }$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 10 \pm 4 } { 2 } }$

$ $ Separate the answer $ $

$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 10 + 4 } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 10 - 4 } { 2 } } \end{array}$

$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 10 } \color{#FF6800}{ + } \color{#FF6800}{ 4 } } { 2 } \\ x = \dfrac { 10 - 4 } { 2 } \end{array}$

$ $ Add $ 10 $ and $ 4$

$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 14 } } { 2 } \\ x = \dfrac { 10 - 4 } { 2 } \end{array}$

$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 14 } { 2 } } \\ x = \dfrac { 10 - 4 } { 2 } \end{array}$

$ $ Do the reduction of the fraction format $ $

$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 7 } { 1 } } \\ x = \dfrac { 10 - 4 } { 2 } \end{array}$

$ $ Reduce the fraction to the lowest term $ $

$\begin{array} {l} x = \color{#FF6800}{ 7 } \\ x = \dfrac { 10 - 4 } { 2 } \end{array}$

$\begin{array} {l} x = 7 \\ x = \dfrac { \color{#FF6800}{ 10 } \color{#FF6800}{ - } \color{#FF6800}{ 4 } } { 2 } \end{array}$

$ $ Subtract $ 4 $ from $ 10$

$\begin{array} {l} x = 7 \\ x = \dfrac { \color{#FF6800}{ 6 } } { 2 } \end{array}$

$\begin{array} {l} x = 7 \\ x = \color{#FF6800}{ \dfrac { 6 } { 2 } } \end{array}$

$ $ Do the reduction of the fraction format $ $

$\begin{array} {l} x = 7 \\ x = \color{#FF6800}{ \dfrac { 3 } { 1 } } \end{array}$

$ $ Reduce the fraction to the lowest term $ $

$\begin{array} {l} x = 7 \\ x = \color{#FF6800}{ 3 } \end{array}$

$ $ 2 real roots $ $

Find the number of solutions

$x ^ { 2 } + 21 = \color{#FF6800}{ 10 } \color{#FF6800}{ x }$

$ $ Move the expression to the left side and change the symbol $ $

$x ^ { 2 } + 21 \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } = 0$

$x ^ { 2 } \color{#FF6800}{ + } \color{#FF6800}{ 21 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } = 0$

$ $ Organize $ x $ in descending power $ $

$x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 21 } = 0$

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 21 } = \color{#FF6800}{ 0 }$

$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$

$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 10 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ 21 }$

$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 10 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 21$

$ $ Remove negative signs because negative numbers raised to even powers are positive $ $

$D = 10 ^ { 2 } - 4 \times 1 \times 21$

$D = \color{#FF6800}{ 10 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 21$

$ $ Calculate power $ $

$D = \color{#FF6800}{ 100 } - 4 \times 1 \times 21$

$D = 100 - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times 21$

$ $ Multiplying any number by 1 does not change the value $ $

$D = 100 - 4 \times 21$

$D = 100 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 21 }$

$ $ Multiply $ - 4 $ and $ 21$

$D = 100 \color{#FF6800}{ - } \color{#FF6800}{ 84 }$

$D = \color{#FF6800}{ 100 } \color{#FF6800}{ - } \color{#FF6800}{ 84 }$

$ $ Subtract $ 84 $ from $ 100$

$D = \color{#FF6800}{ 16 }$

$\color{#FF6800}{ D } = \color{#FF6800}{ 16 }$

$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $

$ $ 2 real roots $ $

$\alpha + \beta = 10 , \alpha \beta = 21$

Find the sum and product of the two roots of the quadratic equation

$x ^ { 2 } + 21 = \color{#FF6800}{ 10 } \color{#FF6800}{ x }$

$ $ Move the expression to the left side and change the symbol $ $

$x ^ { 2 } + 21 \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } = 0$

$x ^ { 2 } \color{#FF6800}{ + } \color{#FF6800}{ 21 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } = 0$

$ $ Organize $ x $ in descending power $ $

$x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 21 } = 0$

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 10 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 21 } = \color{#FF6800}{ 0 }$

$ $ 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}$

$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 10 } { 1 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 21 } { 1 } }$

$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 10 } { 1 } } , \alpha \beta = \dfrac { 21 } { 1 }$

$ $ Solve the sign of a fraction with a negative sign $ $

$\alpha + \beta = \color{#FF6800}{ \dfrac { 10 } { 1 } } , \alpha \beta = \dfrac { 21 } { 1 }$

$\alpha + \beta = \dfrac { 10 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { 21 } { 1 }$

$ $ If the denominator is 1, the denominator can be removed $ $

$\alpha + \beta = \color{#FF6800}{ 10 } , \alpha \beta = \dfrac { 21 } { 1 }$

$\alpha + \beta = 10 , \alpha \beta = \dfrac { 21 } { \color{#FF6800}{ 1 } }$

$ $ If the denominator is 1, the denominator can be removed $ $

$\alpha + \beta = 10 , \alpha \beta = \color{#FF6800}{ 21 }$

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