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Formula
Solve the quadratic equation
Answer
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Number of solution
Answer
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Relationship between roots and coefficients
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Graph
$y = \left ( 40 - x \right ) \left ( 20 - x \right )$
$y = 576$
$x$-intercept
$\left ( 40 , 0 \right )$, $\left ( 20 , 0 \right )$
$y$-intercept
$\left ( 0 , 800 \right )$
Minimum
$\left ( 30 , - 100 \right )$
Standard form
$y = \left ( x - 30 \right ) ^ { 2 } - 100$
$\left( 40-x \right) \left( 20-x \right) = 576$
$\begin{array} {l} x = 56 \\ x = 4 \end{array}$
Find solution by method of factorization
$\left ( 40 - x \right ) \left ( 20 - x \right ) = \color{#FF6800}{ 576 }$
$ $ Move the expression to the left side and change the symbol $ $
$\left ( 40 - x \right ) \left ( 20 - x \right ) - 576 = 0$
$\left ( \color{#FF6800}{ 40 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 20 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \color{#FF6800}{ - } \color{#FF6800}{ 576 } = 0$
$ $ Expand the expression $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 60 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 224 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 60 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 224 } = 0$
$acx^{2} + \left(ad + bc\right)x +bd = \left(ax + b\right)\left(cx+d\right)$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 56 } \right ) \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) = 0$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 56 } \right ) \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \right ) = \color{#FF6800}{ 0 }$
$ $ If the product of the factor is 0, at least one factor should be 0 $ $
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 56 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 4 } = \color{#FF6800}{ 0 } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 56 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 4 } = \color{#FF6800}{ 0 } \end{array}$
$ $ Solve the equation to find $ x$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 56 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 4 } \end{array}$
$\begin{array} {l} x = 56 \\ x = 4 \end{array}$
Solve quadratic equations using the square root
$\left ( \color{#FF6800}{ 40 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 20 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) = 576$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 60 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 800 } = 576$
$x ^ { 2 } - 60 x + 800 = \color{#FF6800}{ 576 }$
$ $ Move the expression to the left side and change the symbol $ $
$x ^ { 2 } - 60 x + 800 \color{#FF6800}{ - } \color{#FF6800}{ 576 } = 0$
$x ^ { 2 } - 60 x + \color{#FF6800}{ 800 } \color{#FF6800}{ - } \color{#FF6800}{ 576 } = 0$
$ $ Subtract $ 576 $ from $ 800$
$x ^ { 2 } - 60 x + \color{#FF6800}{ 224 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 60 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 224 } = \color{#FF6800}{ 0 }$
$ $ Convert the quadratic expression on the left side to a perfect square format $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 30 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 224 } \color{#FF6800}{ - } \color{#FF6800}{ 30 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 30 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 224 } \color{#FF6800}{ - } \color{#FF6800}{ 30 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$ $ Organize the expression $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 30 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 676 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 30 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 676 }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 30 } = \pm \sqrt{ \color{#FF6800}{ 676 } }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 30 } = \pm \sqrt{ \color{#FF6800}{ 676 } }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 26 } \color{#FF6800}{ + } \color{#FF6800}{ 30 }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 26 } \color{#FF6800}{ + } \color{#FF6800}{ 30 }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 30 } \color{#FF6800}{ + } \color{#FF6800}{ 26 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 30 } \color{#FF6800}{ - } \color{#FF6800}{ 26 } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 30 } \color{#FF6800}{ + } \color{#FF6800}{ 26 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 30 } \color{#FF6800}{ - } \color{#FF6800}{ 26 } \end{array}$
$ $ Organize the expression $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 56 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 4 } \end{array}$
$\begin{array} {l} x = 56 \\ x = 4 \end{array}$
Calculate using the quadratic formula
$\left ( \color{#FF6800}{ 40 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 20 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) = 576$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 60 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 800 } = 576$
$x ^ { 2 } - 60 x + 800 = \color{#FF6800}{ 576 }$
$ $ Move the expression to the left side and change the symbol $ $
$x ^ { 2 } - 60 x + 800 \color{#FF6800}{ - } \color{#FF6800}{ 576 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 60 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 800 } \color{#FF6800}{ - } \color{#FF6800}{ 576 } = \color{#FF6800}{ 0 }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 60 \pm \sqrt{ \left ( - 60 \right ) ^ { 2 } - 4 \times 1 \times 224 } } { 2 \times 1 } }$
$x = \dfrac { 60 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 60 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 224 } } { 2 \times 1 }$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$x = \dfrac { 60 \pm \sqrt{ 60 ^ { 2 } - 4 \times 1 \times 224 } } { 2 \times 1 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 60 \pm \sqrt{ 60 ^ { 2 } - 4 \times 1 \times 224 } } { 2 \times 1 } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 60 \pm \sqrt{ 2704 } } { 2 \times 1 } }$
$x = \dfrac { 60 \pm \sqrt{ \color{#FF6800}{ 2704 } } } { 2 \times 1 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { 60 \pm \color{#FF6800}{ 52 } } { 2 \times 1 }$
$x = \dfrac { 60 \pm 52 } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
$ $ Multiplying any number by 1 does not change the value $ $
$x = \dfrac { 60 \pm 52 } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 60 \pm 52 } { 2 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 60 + 52 } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 60 - 52 } { 2 } } \end{array}$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 60 } \color{#FF6800}{ + } \color{#FF6800}{ 52 } } { 2 } \\ x = \dfrac { 60 - 52 } { 2 } \end{array}$
$ $ Add $ 60 $ and $ 52$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 112 } } { 2 } \\ x = \dfrac { 60 - 52 } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 112 } { 2 } } \\ x = \dfrac { 60 - 52 } { 2 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 56 } { 1 } } \\ x = \dfrac { 60 - 52 } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 56 } { 1 } } \\ x = \dfrac { 60 - 52 } { 2 } \end{array}$
$ $ Reduce the fraction to the lowest term $ $
$\begin{array} {l} x = \color{#FF6800}{ 56 } \\ x = \dfrac { 60 - 52 } { 2 } \end{array}$
$\begin{array} {l} x = 56 \\ x = \dfrac { \color{#FF6800}{ 60 } \color{#FF6800}{ - } \color{#FF6800}{ 52 } } { 2 } \end{array}$
$ $ Subtract $ 52 $ from $ 60$
$\begin{array} {l} x = 56 \\ x = \dfrac { \color{#FF6800}{ 8 } } { 2 } \end{array}$
$\begin{array} {l} x = 56 \\ x = \color{#FF6800}{ \dfrac { 8 } { 2 } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = 56 \\ x = \color{#FF6800}{ \dfrac { 4 } { 1 } } \end{array}$
$\begin{array} {l} x = 56 \\ x = \color{#FF6800}{ \dfrac { 4 } { 1 } } \end{array}$
$ $ Reduce the fraction to the lowest term $ $
$\begin{array} {l} x = 56 \\ x = \color{#FF6800}{ 4 } \end{array}$
$ $ 2 real roots $ $
Find the number of solutions
$\left ( \color{#FF6800}{ 40 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 20 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) = 576$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 60 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 800 } = 576$
$x ^ { 2 } - 60 x + 800 = \color{#FF6800}{ 576 }$
$ $ Move the expression to the left side and change the symbol $ $
$x ^ { 2 } - 60 x + 800 \color{#FF6800}{ - } \color{#FF6800}{ 576 } = 0$
$x ^ { 2 } - 60 x + \color{#FF6800}{ 800 } \color{#FF6800}{ - } \color{#FF6800}{ 576 } = 0$
$ $ Subtract $ 576 $ from $ 800$
$x ^ { 2 } - 60 x + \color{#FF6800}{ 224 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 60 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 224 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 60 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ 224 }$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 60 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 224$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$D = 60 ^ { 2 } - 4 \times 1 \times 224$
$D = \color{#FF6800}{ 60 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 224$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 3600 } - 4 \times 1 \times 224$
$D = 3600 - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times 224$
$ $ Multiplying any number by 1 does not change the value $ $
$D = 3600 - 4 \times 224$
$D = 3600 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 224 }$
$ $ Multiply $ - 4 $ and $ 224$
$D = 3600 \color{#FF6800}{ - } \color{#FF6800}{ 896 }$
$D = \color{#FF6800}{ 3600 } \color{#FF6800}{ - } \color{#FF6800}{ 896 }$
$ $ Subtract $ 896 $ from $ 3600$
$D = \color{#FF6800}{ 2704 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 2704 }$
$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $
$ $ 2 real roots $ $
$\alpha + \beta = 60 , \alpha \beta = 224$
Find the sum and product of the two roots of the quadratic equation
$\left ( \color{#FF6800}{ 40 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 20 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) = 576$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 60 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 800 } = 576$
$x ^ { 2 } - 60 x + 800 = \color{#FF6800}{ 576 }$
$ $ Move the expression to the left side and change the symbol $ $
$x ^ { 2 } - 60 x + 800 \color{#FF6800}{ - } \color{#FF6800}{ 576 } = 0$
$x ^ { 2 } - 60 x + \color{#FF6800}{ 800 } \color{#FF6800}{ - } \color{#FF6800}{ 576 } = 0$
$ $ Subtract $ 576 $ from $ 800$
$x ^ { 2 } - 60 x + \color{#FF6800}{ 224 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 60 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 224 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 60 } { 1 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 224 } { 1 } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 60 } { 1 } } , \alpha \beta = \dfrac { 224 } { 1 }$
$ $ Solve the sign of a fraction with a negative sign $ $
$\alpha + \beta = \color{#FF6800}{ \dfrac { 60 } { 1 } } , \alpha \beta = \dfrac { 224 } { 1 }$
$\alpha + \beta = \dfrac { 60 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { 224 } { 1 }$
$ $ If the denominator is 1, the denominator can be removed $ $
$\alpha + \beta = \color{#FF6800}{ 60 } , \alpha \beta = \dfrac { 224 } { 1 }$
$\alpha + \beta = 60 , \alpha \beta = \dfrac { 224 } { \color{#FF6800}{ 1 } }$
$ $ If the denominator is 1, the denominator can be removed $ $
$\alpha + \beta = 60 , \alpha \beta = \color{#FF6800}{ 224 }$
$ $ 그래프 보기 $ $
Graph
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