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Formula

Solve the quadratic equation

Answer

Number of solution

Answer

Relationship between roots and coefficients

Answer

Graph

$y = x ^ { 2 } - 4 x + 1$

$y = 0$

$x$Intercept

$\left ( \sqrt{ 3 } + 2 , 0 \right )$, $\left ( 2 - \sqrt{ 3 } , 0 \right )$

$y$Intercept

$\left ( 0 , 1 \right )$

Minimum

$\left ( 2 , - 3 \right )$

Standard form

$y = \left ( x - 2 \right ) ^ { 2 } - 3$

$x ^{ 2 } -4x+1 = 0$

$\begin{array} {l} x = 2 + \sqrt{ 3 } \\ x = 2 - \sqrt{ 3 } \end{array}$

Solve quadratic equations using the square root

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$

$ $ Convert the quadratic expression on the left side to a perfect square format $ $

$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$

$\left ( x - 2 \right ) ^ { 2 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } = 0$

$ $ Move the constant to the right side and change the sign $ $

$\left ( x - 2 \right ) ^ { 2 } = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } }$

$\left ( x - 2 \right ) ^ { 2 } = - 1 + \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } }$

$ $ Calculate power $ $

$\left ( x - 2 \right ) ^ { 2 } = - 1 + \color{#FF6800}{ 4 }$

$\left ( x - 2 \right ) ^ { 2 } = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 4 }$

$ $ Add $ - 1 $ and $ 4$

$\left ( x - 2 \right ) ^ { 2 } = \color{#FF6800}{ 3 }$

$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 3 }$

$ $ Solve quadratic equations using the square root $ $

$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } = \pm \sqrt{ \color{#FF6800}{ 3 } }$

$ $ Solve a solution to $ x$

$\color{#FF6800}{ x } = \pm \sqrt{ \color{#FF6800}{ 3 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 }$

$ $ Separate the answer $ $

$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 3 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \end{array}$

$\begin{array} {l} x = 2 + \sqrt{ 3 } \\ x = 2 - \sqrt{ 3 } \end{array}$

Calculate using the quadratic formula

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

$ $ Simplify Minus $ $

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

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

$ $ Remove negative signs because negative numbers raised to even powers are positive $ $

$x = \dfrac { 4 \pm \sqrt{ 4 ^ { 2 } - 4 \times 1 \times 1 } } { 2 \times 1 }$

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

$ $ Organize the expression $ $

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 \pm \sqrt{ 12 } } { 2 \times 1 } }$

$x = \dfrac { 4 \pm \sqrt{ \color{#FF6800}{ 12 } } } { 2 \times 1 }$

$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $

$x = \dfrac { 4 \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 3 } } } { 2 \times 1 }$

$x = \dfrac { 4 \pm 2 \sqrt{ 3 } } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$

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

$x = \dfrac { 4 \pm 2 \sqrt{ 3 } } { \color{#FF6800}{ 2 } }$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 \pm 2 \sqrt{ 3 } } { 2 } }$

$ $ Separate the answer $ $

$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 + 2 \sqrt{ 3 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 4 - 2 \sqrt{ 3 } } { 2 } } \end{array}$

$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 4 + 2 \sqrt{ 3 } } { 2 } } \\ x = \dfrac { 4 - 2 \sqrt{ 3 } } { 2 } \end{array}$

$ $ Do the reduction of the fraction format $ $

$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 2 + \sqrt{ 3 } } { 1 } } \\ x = \dfrac { 4 - 2 \sqrt{ 3 } } { 2 } \end{array}$

$\begin{array} {l} x = \dfrac { 2 + \sqrt{ 3 } } { \color{#FF6800}{ 1 } } \\ x = \dfrac { 4 - 2 \sqrt{ 3 } } { 2 } \end{array}$

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

$\begin{array} {l} x = \color{#FF6800}{ 2 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 3 } } \\ x = \dfrac { 4 - 2 \sqrt{ 3 } } { 2 } \end{array}$

$\begin{array} {l} x = 2 + \sqrt{ 3 } \\ x = \color{#FF6800}{ \dfrac { 4 - 2 \sqrt{ 3 } } { 2 } } \end{array}$

$ $ Do the reduction of the fraction format $ $

$\begin{array} {l} x = 2 + \sqrt{ 3 } \\ x = \color{#FF6800}{ \dfrac { 2 - \sqrt{ 3 } } { 1 } } \end{array}$

$\begin{array} {l} x = 2 + \sqrt{ 3 } \\ x = \dfrac { 2 - \sqrt{ 3 } } { \color{#FF6800}{ 1 } } \end{array}$

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

$\begin{array} {l} x = 2 + \sqrt{ 3 } \\ x = \color{#FF6800}{ 2 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 3 } } \end{array}$

$ $ 2 real roots $ $

Find the number of solutions

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$

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

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

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

$ $ Remove negative signs because negative numbers raised to even powers are positive $ $

$D = 4 ^ { 2 } - 4 \times 1 \times 1$

$D = \color{#FF6800}{ 4 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 1$

$ $ Calculate power $ $

$D = \color{#FF6800}{ 16 } - 4 \times 1 \times 1$

$D = 16 - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 }$

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

$D = 16 - 4$

$D = \color{#FF6800}{ 16 } \color{#FF6800}{ - } \color{#FF6800}{ 4 }$

$ $ Subtract $ 4 $ from $ 16$

$D = \color{#FF6800}{ 12 }$

$\color{#FF6800}{ D } = \color{#FF6800}{ 12 }$

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

$ $ 2 real roots $ $

$\alpha + \beta = 4 , \alpha \beta = 1$

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

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$

$ $ 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}$

$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 4 } { 1 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 1 } { 1 } }$

$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 4 } { 1 } } , \alpha \beta = \dfrac { 1 } { 1 }$

$ $ Solve the sign of a fraction with a negative sign $ $

$\alpha + \beta = \color{#FF6800}{ \dfrac { 4 } { 1 } } , \alpha \beta = \dfrac { 1 } { 1 }$

$\alpha + \beta = \dfrac { 4 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { 1 } { 1 }$

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

$\alpha + \beta = \color{#FF6800}{ 4 } , \alpha \beta = \dfrac { 1 } { 1 }$

$\alpha + \beta = 4 , \alpha \beta = \dfrac { 1 } { \color{#FF6800}{ 1 } }$

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

$\alpha + \beta = 4 , \alpha \beta = \color{#FF6800}{ 1 }$

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