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Solve the quadratic equation
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Number of solution
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Relationship between roots and coefficients
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Graph
$y = \left ( 24 - x \right ) \left ( 16 - 2 x \right )$
$y = 210$
$x$Intercept
$\left ( 8 , 0 \right )$, $\left ( 24 , 0 \right )$
$y$Intercept
$\left ( 0 , 384 \right )$
$\left( 24-x \right) \left( 16-2x \right) = 210$
$\begin{array} {l} x = 29 \\ x = 3 \end{array}$
Find solution by method of factorization
$\left ( 24 - x \right ) \left ( 16 - 2 x \right ) = \color{#FF6800}{ 210 }$
$ $ Move the expression to the left side and change the symbol $ $
$\left ( 24 - x \right ) \left ( 16 - 2 x \right ) - 210 = 0$
$\left ( \color{#FF6800}{ 24 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \right ) \color{#FF6800}{ - } \color{#FF6800}{ 210 } = 0$
$ $ Expand the expression $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 64 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 174 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 64 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 174 } = 0$
$acx^{2} + \left(ad + bc\right)x +bd = \left(ax + b\right)\left(cx+d\right)$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 29 } \right ) \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) = 0$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 29 } \right ) \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) = \color{#FF6800}{ 0 }$
$ $ If the product of the factor is 0, at least one factor should be 0 $ $
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 29 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 29 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 } \end{array}$
$ $ Solve the equation to find $ x$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 29 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 3 } \end{array}$
$\begin{array} {l} x = 29 \\ x = 3 \end{array}$
Solve quadratic equations using the square root
$\left ( \color{#FF6800}{ 24 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \right ) = 210$
$ $ Organize the expression $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 64 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 384 } = 210$
$2 x ^ { 2 } - 64 x + 384 = \color{#FF6800}{ 210 }$
$ $ Move the expression to the left side and change the symbol $ $
$2 x ^ { 2 } - 64 x + 384 \color{#FF6800}{ - } \color{#FF6800}{ 210 } = 0$
$2 x ^ { 2 } - 64 x + \color{#FF6800}{ 384 } \color{#FF6800}{ - } \color{#FF6800}{ 210 } = 0$
$ $ Subtract $ 210 $ from $ 384$
$2 x ^ { 2 } - 64 x + \color{#FF6800}{ 174 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 64 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 174 } = \color{#FF6800}{ 0 }$
$ $ Divide both sides by the coefficient of the leading highest term $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 32 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 87 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 32 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 87 } = \color{#FF6800}{ 0 }$
$ $ Convert the quadratic expression on the left side to a perfect square format $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 16 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 87 } \color{#FF6800}{ - } \color{#FF6800}{ 16 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 16 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 87 } \color{#FF6800}{ - } \color{#FF6800}{ 16 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$ $ Organize the expression $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 16 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 169 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 16 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 169 }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 16 } = \pm \sqrt{ \color{#FF6800}{ 169 } }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 16 } = \pm \sqrt{ \color{#FF6800}{ 169 } }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 13 } \color{#FF6800}{ + } \color{#FF6800}{ 16 }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 13 } \color{#FF6800}{ + } \color{#FF6800}{ 16 }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 16 } \color{#FF6800}{ + } \color{#FF6800}{ 13 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ 13 } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 16 } \color{#FF6800}{ + } \color{#FF6800}{ 13 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ 13 } \end{array}$
$ $ Organize the expression $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 29 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 3 } \end{array}$
$\begin{array} {l} x = 29 \\ x = 3 \end{array}$
Calculate using the quadratic formula
$\left ( \color{#FF6800}{ 24 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \right ) = 210$
$ $ Organize the expression $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 64 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 384 } = 210$
$2 x ^ { 2 } - 64 x + 384 = \color{#FF6800}{ 210 }$
$ $ Move the expression to the left side and change the symbol $ $
$2 x ^ { 2 } - 64 x + 384 \color{#FF6800}{ - } \color{#FF6800}{ 210 } = 0$
$2 x ^ { 2 } - 64 x + \color{#FF6800}{ 384 } \color{#FF6800}{ - } \color{#FF6800}{ 210 } = 0$
$ $ Subtract $ 210 $ from $ 384$
$2 x ^ { 2 } - 64 x + \color{#FF6800}{ 174 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 64 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 174 } = 0$
$ $ Bind the expressions with the common factor $ 2$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 32 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 87 } \right ) = 0$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 32 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 87 } \right ) = \color{#FF6800}{ 0 }$
$ $ Divide both sides by $ 2$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 32 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 87 } = \color{#FF6800}{ 0 }$
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 32 \right ) \pm \sqrt{ \left ( - 32 \right ) ^ { 2 } - 4 \times 1 \times 87 } } { 2 \times 1 }$
$ $ Simplify Minus $ $
$x = \dfrac { 32 \pm \sqrt{ \left ( - 32 \right ) ^ { 2 } - 4 \times 1 \times 87 } } { 2 \times 1 }$
$x = \dfrac { 32 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 32 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 87 } } { 2 \times 1 }$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$x = \dfrac { 32 \pm \sqrt{ 32 ^ { 2 } - 4 \times 1 \times 87 } } { 2 \times 1 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 32 \pm \sqrt{ 32 ^ { 2 } - 4 \times 1 \times 87 } } { 2 \times 1 } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 32 \pm \sqrt{ 676 } } { 2 \times 1 } }$
$x = \dfrac { 32 \pm \sqrt{ \color{#FF6800}{ 676 } } } { 2 \times 1 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { 32 \pm \color{#FF6800}{ 26 } } { 2 \times 1 }$
$x = \dfrac { 32 \pm 26 } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
$ $ Multiplying any number by 1 does not change the value $ $
$x = \dfrac { 32 \pm 26 } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 32 \pm 26 } { 2 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 32 + 26 } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 32 - 26 } { 2 } } \end{array}$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 32 } \color{#FF6800}{ + } \color{#FF6800}{ 26 } } { 2 } \\ x = \dfrac { 32 - 26 } { 2 } \end{array}$
$ $ Add $ 32 $ and $ 26$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 58 } } { 2 } \\ x = \dfrac { 32 - 26 } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 58 } { 2 } } \\ x = \dfrac { 32 - 26 } { 2 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 29 } { 1 } } \\ x = \dfrac { 32 - 26 } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 29 } { 1 } } \\ x = \dfrac { 32 - 26 } { 2 } \end{array}$
$ $ Reduce the fraction to the lowest term $ $
$\begin{array} {l} x = \color{#FF6800}{ 29 } \\ x = \dfrac { 32 - 26 } { 2 } \end{array}$
$\begin{array} {l} x = 29 \\ x = \dfrac { \color{#FF6800}{ 32 } \color{#FF6800}{ - } \color{#FF6800}{ 26 } } { 2 } \end{array}$
$ $ Subtract $ 26 $ from $ 32$
$\begin{array} {l} x = 29 \\ x = \dfrac { \color{#FF6800}{ 6 } } { 2 } \end{array}$
$\begin{array} {l} x = 29 \\ x = \color{#FF6800}{ \dfrac { 6 } { 2 } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = 29 \\ x = \color{#FF6800}{ \dfrac { 3 } { 1 } } \end{array}$
$\begin{array} {l} x = 29 \\ x = \color{#FF6800}{ \dfrac { 3 } { 1 } } \end{array}$
$ $ Reduce the fraction to the lowest term $ $
$\begin{array} {l} x = 29 \\ x = \color{#FF6800}{ 3 } \end{array}$
$ $ 2 real roots $ $
Find the number of solutions
$\left ( \color{#FF6800}{ 24 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \right ) = 210$
$ $ Organize the expression $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 64 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 384 } = 210$
$2 x ^ { 2 } - 64 x + 384 = \color{#FF6800}{ 210 }$
$ $ Move the expression to the left side and change the symbol $ $
$2 x ^ { 2 } - 64 x + 384 \color{#FF6800}{ - } \color{#FF6800}{ 210 } = 0$
$2 x ^ { 2 } - 64 x + \color{#FF6800}{ 384 } \color{#FF6800}{ - } \color{#FF6800}{ 210 } = 0$
$ $ Subtract $ 210 $ from $ 384$
$2 x ^ { 2 } - 64 x + \color{#FF6800}{ 174 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 64 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 174 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 64 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 174 }$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 64 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times 174$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$D = 64 ^ { 2 } - 4 \times 2 \times 174$
$D = \color{#FF6800}{ 64 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times 174$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 4096 } - 4 \times 2 \times 174$
$D = 4096 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 174 }$
$ $ Multiply the numbers $ $
$D = 4096 \color{#FF6800}{ - } \color{#FF6800}{ 1392 }$
$D = \color{#FF6800}{ 4096 } \color{#FF6800}{ - } \color{#FF6800}{ 1392 }$
$ $ Subtract $ 1392 $ from $ 4096$
$D = \color{#FF6800}{ 2704 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 2704 }$
$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $
$ $ 2 real roots $ $
$\alpha + \beta = 32 , \alpha \beta = 87$
Find the sum and product of the two roots of the quadratic equation
$\left ( \color{#FF6800}{ 24 } \color{#FF6800}{ - } \color{#FF6800}{ x } \right ) \left ( \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \right ) = 210$
$ $ Organize the expression $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 64 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 384 } = 210$
$2 x ^ { 2 } - 64 x + 384 = \color{#FF6800}{ 210 }$
$ $ Move the expression to the left side and change the symbol $ $
$2 x ^ { 2 } - 64 x + 384 \color{#FF6800}{ - } \color{#FF6800}{ 210 } = 0$
$2 x ^ { 2 } - 64 x + \color{#FF6800}{ 384 } \color{#FF6800}{ - } \color{#FF6800}{ 210 } = 0$
$ $ Subtract $ 210 $ from $ 384$
$2 x ^ { 2 } - 64 x + \color{#FF6800}{ 174 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 64 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 174 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 64 } { 2 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 174 } { 2 } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 64 } { 2 } } , \alpha \beta = \dfrac { 174 } { 2 }$
$ $ Solve the sign of a fraction with a negative sign $ $
$\alpha + \beta = \color{#FF6800}{ \dfrac { 64 } { 2 } } , \alpha \beta = \dfrac { 174 } { 2 }$
$\alpha + \beta = \color{#FF6800}{ \dfrac { 64 } { 2 } } , \alpha \beta = \dfrac { 174 } { 2 }$
$ $ Reduce the fraction $ $
$\alpha + \beta = \color{#FF6800}{ 32 } , \alpha \beta = \dfrac { 174 } { 2 }$
$\alpha + \beta = 32 , \alpha \beta = \color{#FF6800}{ \dfrac { 174 } { 2 } }$
$ $ Reduce the fraction $ $
$\alpha + \beta = 32 , \alpha \beta = \color{#FF6800}{ 87 }$
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