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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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Graph
$y = 2 x ^ { 2 } - 8 x + 16$
$y = 0$
$y$Intercept
$\left ( 0 , 16 \right )$
Minimum
$\left ( 2 , 8 \right )$
Standard form
$y = 2 \left ( x - 2 \right ) ^ { 2 } + 8$
$\begin{array} {l} x = 2 + 2 i \\ x = 2 - 2 i \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 16 } = \color{#FF6800}{ 0 }$
$ $ Divide both sides by the coefficient of the leading highest term $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 8 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 8 } = \color{#FF6800}{ 0 }$
$ $ Convert the quadratic expression on the left side to a perfect square format $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( x - 2 \right ) ^ { 2 } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } = 0$
$ $ Move the constant to the right side and change the sign $ $
$\left ( x - 2 \right ) ^ { 2 } = \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } }$
$\left ( x - 2 \right ) ^ { 2 } = - 8 + \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } }$
$ $ Calculate power $ $
$\left ( x - 2 \right ) ^ { 2 } = - 8 + \color{#FF6800}{ 4 }$
$\left ( x - 2 \right ) ^ { 2 } = \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ + } \color{#FF6800}{ 4 }$
$ $ Add $ - 8 $ and $ 4$
$\left ( x - 2 \right ) ^ { 2 } = \color{#FF6800}{ - } \color{#FF6800}{ 4 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ - } \color{#FF6800}{ 4 }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } = \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 4 } }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } = \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 4 } }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 2 } \color{#FF6800}{ i } \color{#FF6800}{ + } \color{#FF6800}{ 2 }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 2 } \color{#FF6800}{ i } \color{#FF6800}{ + } \color{#FF6800}{ 2 }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ i } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ i } \end{array}$
$\begin{array} {l} x = 2 + 2 i \\ x = 2 - 2 i \end{array}$
Calculate using the quodratic formula$($Imaginary root solution$)
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 16 } = \color{#FF6800}{ 0 }$
$ $ Solve the quadratic equation $ ax^{2}+bx+c=0 $ using the quadratic formula $ \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 8 } \right ) \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 8 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } }$
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 8 \right ) \pm \sqrt{ \left ( - 8 \right ) ^ { 2 } - 4 \times 2 \times 16 } } { 2 \times 2 }$
$ $ Simplify Minus $ $
$x = \dfrac { 8 \pm \sqrt{ \left ( - 8 \right ) ^ { 2 } - 4 \times 2 \times 16 } } { 2 \times 2 }$
$x = \dfrac { 8 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 8 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times 16 } } { 2 \times 2 }$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$x = \dfrac { 8 \pm \sqrt{ 8 ^ { 2 } - 4 \times 2 \times 16 } } { 2 \times 2 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 8 } \pm \sqrt{ \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 8 } \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 64 } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } }$
$x = \dfrac { 8 \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 64 } } } { 2 \times 2 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { 8 \pm \color{#FF6800}{ 8 } \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } { 2 \times 2 }$
$x = \dfrac { 8 \pm 8 \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 1 } } } { 2 \times 2 }$
$ $ It is $ \sqrt{-1} = i$
$x = \dfrac { 8 \pm 8 \color{#FF6800}{ i } } { 2 \times 2 }$
$x = \dfrac { 8 \pm 8 i } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } }$
$ $ Multiply $ 2 $ and $ 2$
$x = \dfrac { 8 \pm 8 i } { \color{#FF6800}{ 4 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 8 } \pm \color{#FF6800}{ 8 } \color{#FF6800}{ i } } { \color{#FF6800}{ 4 } } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 8 } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ i } } { \color{#FF6800}{ 4 } } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 8 } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ i } } { \color{#FF6800}{ 4 } } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 8 } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \color{#FF6800}{ i } } { \color{#FF6800}{ 4 } } } \\ x = \dfrac { 8 - 8 i } { 4 } \end{array}$
$ $ Reduce the fraction $ $
$\begin{array} {l} x = \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ i } \\ x = \dfrac { 8 - 8 i } { 4 } \end{array}$
$\begin{array} {l} x = 2 + 2 i \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 8 } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ i } } { \color{#FF6800}{ 4 } } } \end{array}$
$ $ Reduce the fraction $ $
$\begin{array} {l} x = 2 + 2 i \\ x = \color{#FF6800}{ 2 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \color{#FF6800}{ i } \end{array}$
$ $ Do not have the solution $ $
Calculate using the quadratic formula
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 16 } = \color{#FF6800}{ 0 }$
$ $ Solve the quadratic equation $ ax^{2}+bx+c=0 $ using the quadratic formula $ \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 8 } \right ) \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 8 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } } }$
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 8 \right ) \pm \sqrt{ \left ( - 8 \right ) ^ { 2 } - 4 \times 2 \times 16 } } { 2 \times 2 }$
$ $ Simplify Minus $ $
$x = \dfrac { 8 \pm \sqrt{ \left ( - 8 \right ) ^ { 2 } - 4 \times 2 \times 16 } } { 2 \times 2 }$
$x = \dfrac { 8 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 8 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times 16 } } { 2 \times 2 }$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$x = \dfrac { 8 \pm \sqrt{ 8 ^ { 2 } - 4 \times 2 \times 16 } } { 2 \times 2 }$
$x = \dfrac { 8 \pm \sqrt{ \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times 16 } } { 2 \times 2 }$
$ $ Calculate power $ $
$x = \dfrac { 8 \pm \sqrt{ \color{#FF6800}{ 64 } - 4 \times 2 \times 16 } } { 2 \times 2 }$
$x = \dfrac { 8 \pm \sqrt{ 64 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 } } } { 2 \times 2 }$
$ $ Multiply the numbers $ $
$x = \dfrac { 8 \pm \sqrt{ 64 \color{#FF6800}{ - } \color{#FF6800}{ 128 } } } { 2 \times 2 }$
$x = \dfrac { 8 \pm \sqrt{ \color{#FF6800}{ 64 } \color{#FF6800}{ - } \color{#FF6800}{ 128 } } } { 2 \times 2 }$
$ $ Subtract $ 128 $ from $ 64$
$x = \dfrac { 8 \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 64 } } } { 2 \times 2 }$
$x = \dfrac { 8 \pm \sqrt{ - 64 } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } }$
$ $ Multiply $ 2 $ and $ 2$
$x = \dfrac { 8 \pm \sqrt{ - 64 } } { \color{#FF6800}{ 4 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 8 } \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 64 } } } { \color{#FF6800}{ 4 } } }$
$ $ The square root of a negative number does not exist within the set of real numbers $ $
$ $ Do not have the solution $ $
$ $ Do not have the real root $ $
Find the number of solutions
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 16 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 8 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 }$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 8 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times 16$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$D = 8 ^ { 2 } - 4 \times 2 \times 16$
$D = \color{#FF6800}{ 8 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times 16$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 64 } - 4 \times 2 \times 16$
$D = 64 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 16 }$
$ $ Multiply the numbers $ $
$D = 64 \color{#FF6800}{ - } \color{#FF6800}{ 128 }$
$D = \color{#FF6800}{ 64 } \color{#FF6800}{ - } \color{#FF6800}{ 128 }$
$ $ Subtract $ 128 $ from $ 64$
$D = \color{#FF6800}{ - } \color{#FF6800}{ 64 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ - } \color{#FF6800}{ 64 }$
$ $ Since $ D<0 $ , there is no real root of the following quadratic equation $ $
$ $ Do not have the real root $ $
$\alpha + \beta = 4 , \alpha \beta = 8$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 8 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 16 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } } { \color{#FF6800}{ 2 } } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 16 } } { \color{#FF6800}{ 2 } } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 8 } } { \color{#FF6800}{ 2 } } } , \alpha \beta = \dfrac { 16 } { 2 }$
$ $ Solve the sign of a fraction with a negative sign $ $
$\alpha + \beta = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 8 } } { \color{#FF6800}{ 2 } } } , \alpha \beta = \dfrac { 16 } { 2 }$
$\alpha + \beta = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 8 } } { \color{#FF6800}{ 2 } } } , \alpha \beta = \dfrac { 16 } { 2 }$
$ $ Reduce the fraction $ $
$\alpha + \beta = \color{#FF6800}{ 4 } , \alpha \beta = \dfrac { 16 } { 2 }$
$\alpha + \beta = 4 , \alpha \beta = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 16 } } { \color{#FF6800}{ 2 } } }$
$ $ Reduce the fraction $ $
$\alpha + \beta = 4 , \alpha \beta = \color{#FF6800}{ 8 }$
Solution search results
search-thumbnail-∠ F From the sum ot x^{A}(2)-8 with 3x-12 subtract the sum ofx-9
$∠$ $F$ From the sum $ot$ $x^{A}\left(2\right)-8$ with $3x-12$ subtract the sum $ofx-9$ with $3x-x^{A}\left(2\right)$ Multinlication takes place $\left(2\times A$ $\left(2\right)+5x+21\right)\left(x-$ $5\right)=$ #Please provide solution by using math formula.
10th-13th grade
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search-thumbnail-x^{2}+8x+16=0
$x^{2}+8x+16=0$
10th-13th grade
Algebra
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search-thumbnail-3. x\bar{2+8x}+16=0
$3.$ $x\bar{2+8x} +16=0$ $-$
7th-9th grade
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search-thumbnail-t2 -8x+16=25
$t2$ $-8x+16=25$
10th-13th grade
Algebra
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search-thumbnail-- GIVEN:X+8x+16=0
$-$ GIVEN:X+8x+16=0 SOLUTION
10th-13th grade
Other
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