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Formula

Solve the quadratic equation

Answer

Number of solution

Answer

Relationship between roots and coefficients

Answer

Graph

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

$y = 0$

$x$Intercept

$\left ( - 1 , 0 \right )$

$y$Intercept

$\left ( 0 , 1 \right )$

Minimum

$\left ( - 1 , 0 \right )$

Standard form

$y = \left ( x + 1 \right ) ^ { 2 }$

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

$x = - 1$

Solve quadratic equations using the square root

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

$ $ Express as the perfect square formula $ $

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

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

$ $ Solve quadratic equations using the square root $ $

$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$

$ $ Solve a solution to $ x$

$\color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 1 }$

$x = - 1$

Calculate using the quadratic formula

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

$ $ Calculate power $ $

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

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

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

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

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

$ $ Remove the two numbers if the values are the same and the signs are different $ $

$x = \dfrac { - 2 \pm \sqrt{ 0 } } { 2 \times 1 }$

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

$n square root $ of 0 is 0 $ $

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

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

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

$x = \dfrac { - 2 \pm 0 } { \color{#FF6800}{ 2 } }$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 2 \pm 0 } { 2 } }$

$ $ The value will not be changed even if adding or subtracting 0 $ $

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 2 } { 2 } }$

$x = \color{#FF6800}{ \dfrac { - 2 } { 2 } }$

$ $ Do the reduction of the fraction format $ $

$x = \color{#FF6800}{ \dfrac { - 1 } { 1 } }$

$x = \dfrac { - 1 } { \color{#FF6800}{ 1 } }$

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

$x = \color{#FF6800}{ - } \color{#FF6800}{ 1 }$

$ $ 1 real root (multiple root) $ $

Find the number of solutions

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

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

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

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

$ $ Calculate power $ $

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

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

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

$D = 4 - 4$

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

$ $ 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) $ $

$\alpha + \beta = - 2 , \alpha \beta = 1$

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

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

$ $ 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}$

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

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

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

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

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

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

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

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