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Solve the quadratic equation

Answer

Number of solution

Answer

Relationship between roots and coefficients

Answer

Graph

$y = x ^ { 2 }$

$y = \sqrt{ 2 }$

$x$Intercept

$\left ( 0 , 0 \right )$

$y$Intercept

$\left ( 0 , 0 \right )$

Minimum

$\left ( 0 , 0 \right )$

Standard form

$y = x ^ { 2 }$

$\begin{array} {l} x = \sqrt[ 4 ]{ 2 } \\ x = - \sqrt[ 4 ]{ 2 } \end{array}$

Calculate using the quadratic formula

$x ^ { 2 } = \sqrt{ \color{#FF6800}{ 2 } }$

$ $ Move the expression to the left side and change the symbol $ $

$x ^ { 2 } - \sqrt{ 2 } = 0$

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$

$ $ Solve the quadratic equation $ ax^{2}+bx+c=0 $ using the quadratic formula $ \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 0 } \pm \sqrt{ \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } } }$

$x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 0 } \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times \left ( - \sqrt{ 2 } \right ) } } { 2 \times 1 }$

$ $ 0 has no sign $ $

$x = \dfrac { \color{#FF6800}{ 0 } \pm \sqrt{ 0 ^ { 2 } - 4 \times 1 \times \left ( - \sqrt{ 2 } \right ) } } { 2 \times 1 }$

$x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - \sqrt{ 2 } \right ) } } { 2 \times 1 }$

$ $ The power of 0 is 0 $ $

$x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 0 } - 4 \times 1 \times \left ( - \sqrt{ 2 } \right ) } } { 2 \times 1 }$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 0 } \pm \sqrt{ \color{#FF6800}{ 0 } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right ) } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } } }$

$ $ Organize the expression $ $

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 0 } \pm \sqrt{ \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } } }$

$x = \dfrac { 0 \pm \sqrt{ \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } } } { 2 \times 1 }$

$ $ Calculate the double radical sign $ $

$x = \dfrac { 0 \pm \color{#FF6800}{ 2 } \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } } { 2 \times 1 }$

$x = \dfrac { 0 \pm 2 \sqrt[ 4 ]{ 2 } } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$

$ $ Multiplying any number by 1 does not change the value $ $

$x = \dfrac { 0 \pm 2 \sqrt[ 4 ]{ 2 } } { \color{#FF6800}{ 2 } }$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 0 } \pm \color{#FF6800}{ 2 } \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 2 } } }$

$ $ Separate the answer $ $

$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 0 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 2 } } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 0 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 2 } } } \end{array}$

$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 0 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 2 } } } \\ x = \dfrac { 0 - 2 \sqrt[ 4 ]{ 2 } } { 2 } \end{array}$

$ $ Simplify the fraction $ $

$\begin{array} {l} x = \color{#FF6800}{ 0 } \color{#FF6800}{ + } \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } \\ x = \dfrac { 0 - 2 \sqrt[ 4 ]{ 2 } } { 2 } \end{array}$

$\begin{array} {l} x = \color{#FF6800}{ 0 } + \sqrt[ 4 ]{ 2 } \\ x = \dfrac { 0 - 2 \sqrt[ 4 ]{ 2 } } { 2 } \end{array}$

$ $ 0 does not change when you add or subtract $ $

$\begin{array} {l} x = \sqrt[ 4 ]{ 2 } \\ x = \dfrac { 0 - 2 \sqrt[ 4 ]{ 2 } } { 2 } \end{array}$

$\begin{array} {l} x = \sqrt[ 4 ]{ 2 } \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 0 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 2 } } } \end{array}$

$ $ Simplify the fraction $ $

$\begin{array} {l} x = \sqrt[ 4 ]{ 2 } \\ x = \color{#FF6800}{ 0 } \color{#FF6800}{ - } \sqrt[ \color{#FF6800}{ 4 } ]{ \color{#FF6800}{ 2 } } \end{array}$

$\begin{array} {l} x = \sqrt[ 4 ]{ 2 } \\ x = \color{#FF6800}{ 0 } - \sqrt[ 4 ]{ 2 } \end{array}$

$ $ 0 does not change when you add or subtract $ $

$\begin{array} {l} x = \sqrt[ 4 ]{ 2 } \\ x = - \sqrt[ 4 ]{ 2 } \end{array}$

$ $ 2 real roots $ $

Find the number of solutions

$x ^ { 2 } = \sqrt{ \color{#FF6800}{ 2 } }$

$ $ Move the expression to the left side and change the symbol $ $

$x ^ { 2 } - \sqrt{ 2 } = 0$

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$

$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$

$\color{#FF6800}{ D } = \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right )$

$D = \color{#FF6800}{ 0 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - \sqrt{ 2 } \right )$

$ $ The power of 0 is 0 $ $

$D = \color{#FF6800}{ 0 } - 4 \times 1 \times \left ( - \sqrt{ 2 } \right )$

$ $ 0 does not change when you add or subtract $ $

$D = - 4 \times 1 \times \left ( - \sqrt{ 2 } \right )$

$D = - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times \left ( - \sqrt{ 2 } \right )$

$ $ Multiplying any number by 1 does not change the value $ $

$D = - 4 \times \left ( - \sqrt{ 2 } \right )$

$D = \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \right )$

$ $ Simplify the expression $ $

$D = \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } }$

$\color{#FF6800}{ D } = \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } }$

$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $

$ $ 2 real roots $ $

$\alpha + \beta = 0 , \alpha \beta = - \sqrt{ 2 }$

Find the sum and product of the two roots of the quadratic equation

$x ^ { 2 } = \sqrt{ \color{#FF6800}{ 2 } }$

$ $ Move the expression to the left side and change the symbol $ $

$x ^ { 2 } - \sqrt{ 2 } = 0$

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$

$ $ 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}$

$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 0 } } { \color{#FF6800}{ 1 } } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 1 } } }$

$\alpha + \beta = - \dfrac { 0 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { - \sqrt{ 2 } } { 1 }$

$ $ If the denominator is 1, the denominator can be removed $ $

$\alpha + \beta = - \color{#FF6800}{ 0 } , \alpha \beta = \dfrac { - \sqrt{ 2 } } { 1 }$

$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ 0 } , \alpha \beta = \dfrac { - \sqrt{ 2 } } { 1 }$

$ $ 0 has no sign $ $

$\alpha + \beta = \color{#FF6800}{ 0 } , \alpha \beta = \dfrac { - \sqrt{ 2 } } { 1 }$

$\alpha + \beta = 0 , \alpha \beta = \dfrac { - \sqrt{ 2 } } { \color{#FF6800}{ 1 } }$

$ $ If the denominator is 1, the denominator can be removed $ $

$\alpha + \beta = 0 , \alpha \beta = \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } }$

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