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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
$y = 2 x ^ { 2 } + 4 x + 3$
$y = 0$
$y$Intercept
$\left ( 0 , 3 \right )$
Minimum
$\left ( - 1 , 1 \right )$
Standard form
$y = 2 \left ( x + 1 \right ) ^ { 2 } + 1$
$2x ^{ 2 } +4x+3 = 0$
$\begin{array} {l} x = \dfrac { - 2 + \sqrt{ 2 } i } { 2 } \\ x = \dfrac { - 2 - \sqrt{ 2 } i } { 2 } \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 }$
$ $ Divide both sides by the coefficient of the leading highest term $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 } { 2 } } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 } { 2 } } = \color{#FF6800}{ 0 }$
$ $ Convert the quadratic expression on the left side to a perfect square format $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 3 } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$ $ Organize the expression $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 2 } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 2 } }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 2 } } }$
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 2 } } }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { \sqrt{ 2 } i } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { \sqrt{ 2 } i } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \sqrt{ 2 } i } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \sqrt{ 2 } i } { 2 } } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { \sqrt{ 2 } i } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \sqrt{ 2 } i } { 2 } } \end{array}$
$ $ Organize the expression $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 2 + \sqrt{ 2 } i } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 2 - \sqrt{ 2 } i } { 2 } } \end{array}$
$\begin{array} {l} x = \dfrac { - 2 + \sqrt{ 2 } i } { 2 } \\ x = \dfrac { - 2 - \sqrt{ 2 } i } { 2 } \end{array}$
Calculate using the quodratic formula$($Imaginary root solution$)
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 }$
$ $ Solve the quadratic equation $ ax^{2}+bx+c=0 $ using the quadratic formula $ \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 4 \pm \sqrt{ 4 ^ { 2 } - 4 \times 2 \times 3 } } { 2 \times 2 } }$
$x = \dfrac { - 4 \pm \sqrt{ \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times 3 } } { 2 \times 2 }$
$ $ Calculate power $ $
$x = \dfrac { - 4 \pm \sqrt{ \color{#FF6800}{ 16 } - 4 \times 2 \times 3 } } { 2 \times 2 }$
$x = \dfrac { - 4 \pm \sqrt{ 16 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } } } { 2 \times 2 }$
$ $ Multiply the numbers $ $
$x = \dfrac { - 4 \pm \sqrt{ 16 \color{#FF6800}{ - } \color{#FF6800}{ 24 } } } { 2 \times 2 }$
$x = \dfrac { - 4 \pm \sqrt{ \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ 24 } } } { 2 \times 2 }$
$ $ Subtract $ 24 $ from $ 16$
$x = \dfrac { - 4 \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 8 } } } { 2 \times 2 }$
$x = \dfrac { - 4 \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 8 } } } { 2 \times 2 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { - 4 \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ i } } { 2 \times 2 }$
$x = \dfrac { - 4 \pm 2 \sqrt{ 2 } i } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } }$
$ $ Multiply $ 2 $ and $ 2$
$x = \dfrac { - 4 \pm 2 \sqrt{ 2 } i } { \color{#FF6800}{ 4 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 4 \pm 2 \sqrt{ 2 } i } { 4 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 4 + 2 \sqrt{ 2 } i } { 4 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 4 - 2 \sqrt{ 2 } i } { 4 } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { - 4 + 2 \sqrt{ 2 } i } { 4 } } \\ x = \dfrac { - 4 - 2 \sqrt{ 2 } i } { 4 } \end{array}$
$ $ Reduce the fraction $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { - 2 + \sqrt{ 2 } i } { 2 } } \\ x = \dfrac { - 4 - 2 \sqrt{ 2 } i } { 4 } \end{array}$
$\begin{array} {l} x = \dfrac { - 2 + \sqrt{ 2 } i } { 2 } \\ x = \color{#FF6800}{ \dfrac { - 4 - 2 \sqrt{ 2 } i } { 4 } } \end{array}$
$ $ Reduce the fraction $ $
$\begin{array} {l} x = \dfrac { - 2 + \sqrt{ 2 } i } { 2 } \\ x = \color{#FF6800}{ \dfrac { - 2 - \sqrt{ 2 } i } { 2 } } \end{array}$
$ $ Do not have the solution $ $
Calculate using the quadratic formula
$x = \dfrac { - 4 \pm \sqrt{ \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times 3 } } { 2 \times 2 }$
$ $ Calculate power $ $
$x = \dfrac { - 4 \pm \sqrt{ \color{#FF6800}{ 16 } - 4 \times 2 \times 3 } } { 2 \times 2 }$
$x = \dfrac { - 4 \pm \sqrt{ 16 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } } } { 2 \times 2 }$
$ $ Multiply the numbers $ $
$x = \dfrac { - 4 \pm \sqrt{ 16 \color{#FF6800}{ - } \color{#FF6800}{ 24 } } } { 2 \times 2 }$
$x = \dfrac { - 4 \pm \sqrt{ \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ 24 } } } { 2 \times 2 }$
$ $ Subtract $ 24 $ from $ 16$
$x = \dfrac { - 4 \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 8 } } } { 2 \times 2 }$
$x = \dfrac { - 4 \pm \sqrt{ - 8 } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } }$
$ $ Multiply $ 2 $ and $ 2$
$x = \dfrac { - 4 \pm \sqrt{ - 8 } } { \color{#FF6800}{ 4 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 4 \pm \sqrt{ - 8 } } { 4 } }$
$ $ 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
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 }$
$D = \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times 3$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 16 } - 4 \times 2 \times 3$
$D = 16 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 }$
$ $ Multiply the numbers $ $
$D = 16 \color{#FF6800}{ - } \color{#FF6800}{ 24 }$
$D = \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ 24 }$
$ $ Subtract $ 24 $ from $ 16$
$D = \color{#FF6800}{ - } \color{#FF6800}{ 8 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ - } \color{#FF6800}{ 8 }$
$ $ Since $ D<0 $ , there is no real root of the following quadratic equation $ $
$ $ Do not have the real root $ $
$\alpha + \beta = - 2 , \alpha \beta = \dfrac { 3 } { 2 }$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 4 } { 2 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 3 } { 2 } }$
$\alpha + \beta = - \color{#FF6800}{ \dfrac { 4 } { 2 } } , \alpha \beta = \dfrac { 3 } { 2 }$
$ $ Reduce the fraction $ $
$\alpha + \beta = - \color{#FF6800}{ 2 } , \alpha \beta = \dfrac { 3 } { 2 }$
Solution search results
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$∠$ $F$ 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.
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$αa^{2}+yx+3=0$
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$2x^{2}$ $4x+3=0$
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$1\right)$ $x^{2}+4x+3=0$ $2$
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$x^{2+4x+3}=0$ $x^{2-5x+4}=0$ $x^{2}-49=0$ $4x^{2-25}=0$ $2x^{2+7x+3}=0$
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search-thumbnail-\sqrt{5}x^{1}-2\sqrt{2}x-2\sqrt{3}=
$x^{2}+4x+3=0$ $\right)$ ) $2x^{2}+4x+5=0$ $\sqrt{5} x^{1}-2\sqrt{2} x-2\sqrt{3} =$ wf fs feafefarn frum
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