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Solve the quadratic equation
Answer
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Number of solution
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Relationship between roots and coefficients
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Solve the equation
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Graph
$y = \dfrac { x \left ( x - 3 \right ) } { 2 }$
$y = 54$
$x$Intercept
$\left ( 3 , 0 \right )$, $\left ( 0 , 0 \right )$
$y$Intercept
$\left ( 0 , 0 \right )$
Minimum
$\left ( \dfrac { 3 } { 2 } , - \dfrac { 9 } { 8 } \right )$
Standard form
$y = \dfrac { 1 } { 2 } \left ( x - \dfrac { 3 } { 2 } \right ) ^ { 2 } - \dfrac { 9 } { 8 }$
$\dfrac{ x \left( x-3 \right) }{ 2 } = 54$
$\begin{array} {l} x = 12 \\ x = - 9 \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ \dfrac { x \left ( x - 3 \right ) } { 2 } } = 54$
$ $ Arrange the fraction expression $ $
$\color{#FF6800}{ \dfrac { x ^ { 2 } - 3 x } { 2 } } = 54$
$\color{#FF6800}{ \dfrac { x ^ { 2 } - 3 x } { 2 } } = \color{#FF6800}{ 54 }$
$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ x } = \color{#FF6800}{ 108 }$
$x ^ { 2 } - 3 x = \color{#FF6800}{ 108 }$
$ $ Move the expression to the left side and change the symbol $ $
$x ^ { 2 } - 3 x \color{#FF6800}{ - } \color{#FF6800}{ 108 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 108 } = \color{#FF6800}{ 0 }$
$ $ Convert the quadratic expression on the left side to a perfect square format $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 108 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 3 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( x - \dfrac { 3 } { 2 } \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 108 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 3 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
$ $ Move the constant to the right side and change the sign $ $
$\left ( x - \dfrac { 3 } { 2 } \right ) ^ { 2 } = \color{#FF6800}{ 108 } \color{#FF6800}{ + } \left ( \color{#FF6800}{ \dfrac { 3 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } }$
$\left ( x - \dfrac { 3 } { 2 } \right ) ^ { 2 } = 108 + \left ( \color{#FF6800}{ \dfrac { 3 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } }$
$ $ When raising a fraction to the power, raise the numerator and denominator each to the power $ $
$\left ( x - \dfrac { 3 } { 2 } \right ) ^ { 2 } = 108 + \dfrac { \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 108 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 ^ { 2 } } { 2 ^ { 2 } } }$
$ $ Organize the expression $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 441 } { 4 } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 441 } { 4 } }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 2 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 441 } { 4 } } }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 3 } { 2 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 441 } { 4 } } }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { 21 } { 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 } { 2 } }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { 21 } { 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 } { 2 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 3 } { 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 21 } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 3 } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 21 } { 2 } } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 3 } { 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 21 } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 3 } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 21 } { 2 } } \end{array}$
$ $ Organize the expression $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 12 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 9 } \end{array}$
$\begin{array} {l} x = 12 \\ x = - 9 \end{array}$
Calculate using the quadratic formula
$\color{#FF6800}{ \dfrac { x \left ( x - 3 \right ) } { 2 } } = 54$
$ $ Arrange the fraction expression $ $
$\color{#FF6800}{ \dfrac { x ^ { 2 } - 3 x } { 2 } } = 54$
$\color{#FF6800}{ \dfrac { x ^ { 2 } - 3 x } { 2 } } = \color{#FF6800}{ 54 }$
$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ x } = \color{#FF6800}{ 108 }$
$x ^ { 2 } - 3 x = \color{#FF6800}{ 108 }$
$ $ Move the expression to the left side and change the symbol $ $
$x ^ { 2 } - 3 x \color{#FF6800}{ - } \color{#FF6800}{ 108 } = 0$
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 3 \right ) \pm \sqrt{ \left ( - 3 \right ) ^ { 2 } - 4 \times 1 \times \left ( - 108 \right ) } } { 2 \times 1 }$
$ $ Simplify Minus $ $
$x = \dfrac { 3 \pm \sqrt{ \left ( - 3 \right ) ^ { 2 } - 4 \times 1 \times \left ( - 108 \right ) } } { 2 \times 1 }$
$x = \dfrac { 3 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - 108 \right ) } } { 2 \times 1 }$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$x = \dfrac { 3 \pm \sqrt{ 3 ^ { 2 } - 4 \times 1 \times \left ( - 108 \right ) } } { 2 \times 1 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 3 \pm \sqrt{ 3 ^ { 2 } - 4 \times 1 \times \left ( - 108 \right ) } } { 2 \times 1 } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 3 \pm \sqrt{ 441 } } { 2 \times 1 } }$
$x = \dfrac { 3 \pm \sqrt{ \color{#FF6800}{ 441 } } } { 2 \times 1 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { 3 \pm \color{#FF6800}{ 21 } } { 2 \times 1 }$
$x = \dfrac { 3 \pm 21 } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
$ $ Multiplying any number by 1 does not change the value $ $
$x = \dfrac { 3 \pm 21 } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 3 \pm 21 } { 2 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 3 + 21 } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 3 - 21 } { 2 } } \end{array}$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ + } \color{#FF6800}{ 21 } } { 2 } \\ x = \dfrac { 3 - 21 } { 2 } \end{array}$
$ $ Add $ 3 $ and $ 21$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 24 } } { 2 } \\ x = \dfrac { 3 - 21 } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 24 } { 2 } } \\ x = \dfrac { 3 - 21 } { 2 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 12 } { 1 } } \\ x = \dfrac { 3 - 21 } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 12 } { 1 } } \\ x = \dfrac { 3 - 21 } { 2 } \end{array}$
$ $ Reduce the fraction to the lowest term $ $
$\begin{array} {l} x = \color{#FF6800}{ 12 } \\ x = \dfrac { 3 - 21 } { 2 } \end{array}$
$\begin{array} {l} x = 12 \\ x = \dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ - } \color{#FF6800}{ 21 } } { 2 } \end{array}$
$ $ Subtract $ 21 $ from $ 3$
$\begin{array} {l} x = 12 \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 18 } } { 2 } \end{array}$
$\begin{array} {l} x = 12 \\ x = \color{#FF6800}{ \dfrac { - 18 } { 2 } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = 12 \\ x = \color{#FF6800}{ \dfrac { - 9 } { 1 } } \end{array}$
$\begin{array} {l} x = 12 \\ x = \dfrac { - 9 } { \color{#FF6800}{ 1 } } \end{array}$
$ $ If the denominator is 1, the denominator can be removed $ $
$\begin{array} {l} x = 12 \\ x = \color{#FF6800}{ - } \color{#FF6800}{ 9 } \end{array}$
$ $ 2 real roots $ $
Find the number of solutions
$\color{#FF6800}{ \dfrac { x \left ( x - 3 \right ) } { 2 } } = 54$
$ $ Arrange the fraction expression $ $
$\color{#FF6800}{ \dfrac { x ^ { 2 } - 3 x } { 2 } } = 54$
$\color{#FF6800}{ \dfrac { x ^ { 2 } - 3 x } { 2 } } = \color{#FF6800}{ 54 }$
$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ x } = \color{#FF6800}{ 108 }$
$x ^ { 2 } - 3 x = \color{#FF6800}{ 108 }$
$ $ Move the expression to the left side and change the symbol $ $
$x ^ { 2 } - 3 x \color{#FF6800}{ - } \color{#FF6800}{ 108 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 108 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 108 } \right )$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - 108 \right )$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$D = 3 ^ { 2 } - 4 \times 1 \times \left ( - 108 \right )$
$D = \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - 108 \right )$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 9 } - 4 \times 1 \times \left ( - 108 \right )$
$D = 9 - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times \left ( - 108 \right )$
$ $ Multiplying any number by 1 does not change the value $ $
$D = 9 - 4 \times \left ( - 108 \right )$
$D = 9 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 108 } \right )$
$ $ Multiply $ - 4 $ and $ - 108$
$D = 9 + \color{#FF6800}{ 432 }$
$D = \color{#FF6800}{ 9 } \color{#FF6800}{ + } \color{#FF6800}{ 432 }$
$ $ Add $ 9 $ and $ 432$
$D = \color{#FF6800}{ 441 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 441 }$
$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $
$ $ 2 real roots $ $
$\alpha + \beta = 3 , \alpha \beta = - 108$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ \dfrac { x \left ( x - 3 \right ) } { 2 } } = 54$
$ $ Arrange the fraction expression $ $
$\color{#FF6800}{ \dfrac { x ^ { 2 } - 3 x } { 2 } } = 54$
$\color{#FF6800}{ \dfrac { x ^ { 2 } - 3 x } { 2 } } = \color{#FF6800}{ 54 }$
$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ x } = \color{#FF6800}{ 108 }$
$x ^ { 2 } - 3 x = \color{#FF6800}{ 108 }$
$ $ Move the expression to the left side and change the symbol $ $
$x ^ { 2 } - 3 x \color{#FF6800}{ - } \color{#FF6800}{ 108 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 108 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 3 } { 1 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 108 } { 1 } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 3 } { 1 } } , \alpha \beta = \dfrac { - 108 } { 1 }$
$ $ Solve the sign of a fraction with a negative sign $ $
$\alpha + \beta = \color{#FF6800}{ \dfrac { 3 } { 1 } } , \alpha \beta = \dfrac { - 108 } { 1 }$
$\alpha + \beta = \dfrac { 3 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { - 108 } { 1 }$
$ $ If the denominator is 1, the denominator can be removed $ $
$\alpha + \beta = \color{#FF6800}{ 3 } , \alpha \beta = \dfrac { - 108 } { 1 }$
$\alpha + \beta = 3 , \alpha \beta = \dfrac { - 108 } { \color{#FF6800}{ 1 } }$
$ $ If the denominator is 1, the denominator can be removed $ $
$\alpha + \beta = 3 , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ 108 }$
$x = 12 \\ x = - 9$
Solve the fractional equation
$\color{#FF6800}{ \dfrac { x \left ( x - 3 \right ) } { 2 } } = \color{#FF6800}{ 54 }$
$ $ If $ \frac{a(x)}{b(x)} = c(x) $ is valid, it is $ \begin{cases} a(x) = b(x) c(x) \\ b(x) \ne 0 \end{cases}$
$\begin{cases} \color{#FF6800}{ x } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) = \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 54 } \\ \color{#FF6800}{ 2 } \neq \color{#FF6800}{ 0 } \end{cases}$
$\begin{cases} \color{#FF6800}{ x } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) = \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 54 } \\ \color{#FF6800}{ 2 } \neq \color{#FF6800}{ 0 } \end{cases}$
$ $ Simplify the expression $ $
$\begin{cases} \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ x } = \color{#FF6800}{ 108 } \\ \color{#FF6800}{ 2 } \neq \color{#FF6800}{ 0 } \end{cases}$
$\begin{cases} \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \color{#FF6800}{ x } = \color{#FF6800}{ 108 } \\ 2 \neq 0 \end{cases}$
$ $ Find solution by method of factorization $ $
$\begin{cases} \begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 12 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 9 } \end{array} \\ 2 \neq 0 \end{cases}$
$\begin{cases} \begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 12 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 9 } \end{array} \\ \color{#FF6800}{ 2 } \neq \color{#FF6800}{ 0 } \end{cases}$
$ $ Substitute the exponential inequality $ $
$\begin{cases} \color{#FF6800}{ x } = \color{#FF6800}{ 12 } \\ \color{#FF6800}{ 2 } \neq \color{#FF6800}{ 0 } \end{cases} \\ \begin{cases} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 9 } \\ \color{#FF6800}{ 2 } \neq \color{#FF6800}{ 0 } \end{cases}$
$\begin{cases} x = 12 \\ \color{#FF6800}{ 2 } \neq \color{#FF6800}{ 0 } \end{cases} \\ \begin{cases} x = - 9 \\ 2 \neq 0 \end{cases}$
$ $ There are infinitely many solutions if both sides of $ \ne $ are different. $ $
$\begin{cases} x = 12 \\ \text{해가 무수히 많습니다} \end{cases} \\ \begin{cases} x = - 9 \\ 2 \neq 0 \end{cases}$
$\begin{cases} \color{#FF6800}{ x } = \color{#FF6800}{ 12 } \\ \text{해가 무수히 많습니다} \end{cases} \\ \begin{cases} x = - 9 \\ 2 \neq 0 \end{cases}$
$ $ Ignore the cases where the system of equations where there are infinitely many solutions. $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ 12 } \\ \begin{cases} x = - 9 \\ 2 \neq 0 \end{cases}$
$x = 12 \\ \begin{cases} x = - 9 \\ \color{#FF6800}{ 2 } \neq \color{#FF6800}{ 0 } \end{cases}$
$ $ There are infinitely many solutions if both sides of $ \ne $ are different. $ $
$x = 12 \\ \begin{cases} x = - 9 \\ \text{해가 무수히 많습니다} \end{cases}$
$x = 12 \\ \begin{cases} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 9 } \\ \text{해가 무수히 많습니다} \end{cases}$
$ $ Ignore the cases where the system of equations where there are infinitely many solutions. $ $
$x = 12 \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 9 }$
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