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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
Answer
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Graph
$y = 2 x ^ { 2 } - 12 x + 18$
$y = 0$
$x$Intercept
$\left ( 3 , 0 \right )$
$y$Intercept
$\left ( 0 , 18 \right )$
Minimum
$\left ( 3 , 0 \right )$
Standard form
$y = 2 \left ( x - 3 \right ) ^ { 2 }$
$2x ^{ 2 } -12x+18 = 0$
$x = 3$
Solve quadratic equations using the square root
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 18 } = \color{#FF6800}{ 0 }$
$ $ Divide both sides by the coefficient of the leading highest term $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } = 0$
$ $ Express as the perfect square formula $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \color{#FF6800}{ 3 }$
$x = 3$
Calculate using the quadratic formula
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 18 } = 0$
$ $ Bind the expressions with the common factor $ 2$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } \right ) = 0$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } \right ) = \color{#FF6800}{ 0 }$
$ $ Divide both sides by $ 2$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 9 } = \color{#FF6800}{ 0 }$
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 6 \right ) \pm \sqrt{ \left ( - 6 \right ) ^ { 2 } - 4 \times 1 \times 9 } } { 2 \times 1 }$
$ $ Simplify Minus $ $
$x = \dfrac { 6 \pm \sqrt{ \left ( - 6 \right ) ^ { 2 } - 4 \times 1 \times 9 } } { 2 \times 1 }$
$x = \dfrac { 6 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 6 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 9 } } { 2 \times 1 }$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$x = \dfrac { 6 \pm \sqrt{ 6 ^ { 2 } - 4 \times 1 \times 9 } } { 2 \times 1 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm \sqrt{ 6 ^ { 2 } - 4 \times 1 \times 9 } } { 2 \times 1 } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm \sqrt{ 0 } } { 2 \times 1 } }$
$x = \dfrac { 6 \pm \sqrt{ \color{#FF6800}{ 0 } } } { 2 \times 1 }$
$n square root $ of 0 is 0 $ $
$x = \dfrac { 6 \pm \color{#FF6800}{ 0 } } { 2 \times 1 }$
$x = \dfrac { 6 \pm 0 } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
$ $ Multiplying any number by 1 does not change the value $ $
$x = \dfrac { 6 \pm 0 } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm 0 } { 2 } }$
$ $ The value will not be changed even if adding or subtracting 0 $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 } { 2 } }$
$x = \color{#FF6800}{ \dfrac { 6 } { 2 } }$
$ $ Do the reduction of the fraction format $ $
$x = \color{#FF6800}{ \dfrac { 3 } { 1 } }$
$x = \color{#FF6800}{ \dfrac { 3 } { 1 } }$
$ $ Reduce the fraction to the lowest term $ $
$x = \color{#FF6800}{ 3 }$
$ $ 1 real root (multiple root) $ $
Find the number of solutions
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 18 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 12 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 18 }$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 12 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times 18$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$D = 12 ^ { 2 } - 4 \times 2 \times 18$
$D = \color{#FF6800}{ 12 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times 18$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 144 } - 4 \times 2 \times 18$
$D = 144 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 18 }$
$ $ Multiply the numbers $ $
$D = 144 \color{#FF6800}{ - } \color{#FF6800}{ 144 }$
$D = \color{#FF6800}{ 144 } \color{#FF6800}{ - } \color{#FF6800}{ 144 }$
$ $ Remove the two numbers if the values are the same and the signs are different $ $
$D = 0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 0 }$
$ $ Since $ D=0 $ , the number of real root of the following quadratic equation is 1 (multiple root) $ $
$ $ 1 real root (multiple root) $ $
$\alpha + \beta = 6 , \alpha \beta = 9$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 12 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 18 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 12 } { 2 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 18 } { 2 } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 12 } { 2 } } , \alpha \beta = \dfrac { 18 } { 2 }$
$ $ Solve the sign of a fraction with a negative sign $ $
$\alpha + \beta = \color{#FF6800}{ \dfrac { 12 } { 2 } } , \alpha \beta = \dfrac { 18 } { 2 }$
$\alpha + \beta = \color{#FF6800}{ \dfrac { 12 } { 2 } } , \alpha \beta = \dfrac { 18 } { 2 }$
$ $ Reduce the fraction $ $
$\alpha + \beta = \color{#FF6800}{ 6 } , \alpha \beta = \dfrac { 18 } { 2 }$
$\alpha + \beta = 6 , \alpha \beta = \color{#FF6800}{ \dfrac { 18 } { 2 } }$
$ $ Reduce the fraction $ $
$\alpha + \beta = 6 , \alpha \beta = \color{#FF6800}{ 9 }$
$ $ 그래프 보기 $ $
Graph
Solution search results
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$\left(1\right)$ $7x^{2}-12x+18=0$ a a hae 
$⊂$
10th-13th grade
Algebra
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