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Solve the quadratic equation
Answer
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Number of solution
Answer
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Relationship between roots and coefficients
Answer
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Graph
$y = x ^ { 2 } - 6 x + 1$
$y = 0$
$x$Intercept
$\left ( 2 \sqrt{ 2 } + 3 , 0 \right )$, $\left ( 3 - 2 \sqrt{ 2 } , 0 \right )$
$y$Intercept
$\left ( 0 , 1 \right )$
Minimum
$\left ( 3 , - 8 \right )$
Standard form
$y = \left ( x - 3 \right ) ^ { 2 } - 8$
$x ^{ 2 } -6x+1 = 0$
$\begin{array} {l} x = 3 + 2 \sqrt{ 2 } \\ x = 3 - 2 \sqrt{ 2 } \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$
$ $ Convert the quadratic expression on the left side to a perfect square format $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( x - 3 \right ) ^ { 2 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } = 0$
$ $ Move the constant to the right side and change the sign $ $
$\left ( x - 3 \right ) ^ { 2 } = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } }$
$\left ( x - 3 \right ) ^ { 2 } = - 1 + \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } }$
$ $ Calculate power $ $
$\left ( x - 3 \right ) ^ { 2 } = - 1 + \color{#FF6800}{ 9 }$
$\left ( x - 3 \right ) ^ { 2 } = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 9 }$
$ $ Add $ - 1 $ and $ 9$
$\left ( x - 3 \right ) ^ { 2 } = \color{#FF6800}{ 8 }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 8 }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \pm \sqrt{ \color{#FF6800}{ 8 } }$
$\color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } = \pm \sqrt{ \color{#FF6800}{ 8 } }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 3 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ 3 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \end{array}$
$\begin{array} {l} x = 3 + 2 \sqrt{ 2 } \\ x = 3 - 2 \sqrt{ 2 } \end{array}$
Calculate using the quadratic formula
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 6 \right ) \pm \sqrt{ \left ( - 6 \right ) ^ { 2 } - 4 \times 1 \times 1 } } { 2 \times 1 }$
$ $ Simplify Minus $ $
$x = \dfrac { 6 \pm \sqrt{ \left ( - 6 \right ) ^ { 2 } - 4 \times 1 \times 1 } } { 2 \times 1 }$
$x = \dfrac { 6 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 6 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 1 } } { 2 \times 1 }$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$x = \dfrac { 6 \pm \sqrt{ 6 ^ { 2 } - 4 \times 1 \times 1 } } { 2 \times 1 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm \sqrt{ 6 ^ { 2 } - 4 \times 1 \times 1 } } { 2 \times 1 } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm \sqrt{ 32 } } { 2 \times 1 } }$
$x = \dfrac { 6 \pm \sqrt{ \color{#FF6800}{ 32 } } } { 2 \times 1 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { 6 \pm \color{#FF6800}{ 4 } \sqrt{ \color{#FF6800}{ 2 } } } { 2 \times 1 }$
$x = \dfrac { 6 \pm 4 \sqrt{ 2 } } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
$ $ Multiplying any number by 1 does not change the value $ $
$x = \dfrac { 6 \pm 4 \sqrt{ 2 } } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 \pm 4 \sqrt{ 2 } } { 2 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 + 4 \sqrt{ 2 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { 6 - 4 \sqrt{ 2 } } { 2 } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 6 + 4 \sqrt{ 2 } } { 2 } } \\ x = \dfrac { 6 - 4 \sqrt{ 2 } } { 2 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 3 + 2 \sqrt{ 2 } } { 1 } } \\ x = \dfrac { 6 - 4 \sqrt{ 2 } } { 2 } \end{array}$
$\begin{array} {l} x = \dfrac { 3 + 2 \sqrt{ 2 } } { \color{#FF6800}{ 1 } } \\ x = \dfrac { 6 - 4 \sqrt{ 2 } } { 2 } \end{array}$
$ $ If the denominator is 1, the denominator can be removed $ $
$\begin{array} {l} x = \color{#FF6800}{ 3 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \\ x = \dfrac { 6 - 4 \sqrt{ 2 } } { 2 } \end{array}$
$\begin{array} {l} x = 3 + 2 \sqrt{ 2 } \\ x = \color{#FF6800}{ \dfrac { 6 - 4 \sqrt{ 2 } } { 2 } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = 3 + 2 \sqrt{ 2 } \\ x = \color{#FF6800}{ \dfrac { 3 - 2 \sqrt{ 2 } } { 1 } } \end{array}$
$\begin{array} {l} x = 3 + 2 \sqrt{ 2 } \\ x = \dfrac { 3 - 2 \sqrt{ 2 } } { \color{#FF6800}{ 1 } } \end{array}$
$ $ If the denominator is 1, the denominator can be removed $ $
$\begin{array} {l} x = 3 + 2 \sqrt{ 2 } \\ x = \color{#FF6800}{ 3 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } \end{array}$
$ $ 2 real roots $ $
Find the number of solutions
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 6 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 }$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 6 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 1$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$D = 6 ^ { 2 } - 4 \times 1 \times 1$
$D = \color{#FF6800}{ 6 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 1$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 36 } - 4 \times 1 \times 1$
$D = 36 - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 }$
$ $ Multiplying any number by 1 does not change the value $ $
$D = 36 - 4$
$D = \color{#FF6800}{ 36 } \color{#FF6800}{ - } \color{#FF6800}{ 4 }$
$ $ Subtract $ 4 $ from $ 36$
$D = \color{#FF6800}{ 32 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 32 }$
$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $
$ $ 2 real roots $ $
$\alpha + \beta = 6 , \alpha \beta = 1$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 6 } { 1 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { 1 } { 1 } }$
$\alpha + \beta = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { - 6 } { 1 } } , \alpha \beta = \dfrac { 1 } { 1 }$
$ $ Solve the sign of a fraction with a negative sign $ $
$\alpha + \beta = \color{#FF6800}{ \dfrac { 6 } { 1 } } , \alpha \beta = \dfrac { 1 } { 1 }$
$\alpha + \beta = \dfrac { 6 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { 1 } { 1 }$
$ $ If the denominator is 1, the denominator can be removed $ $
$\alpha + \beta = \color{#FF6800}{ 6 } , \alpha \beta = \dfrac { 1 } { 1 }$
$\alpha + \beta = 6 , \alpha \beta = \dfrac { 1 } { \color{#FF6800}{ 1 } }$
$ $ If the denominator is 1, the denominator can be removed $ $
$\alpha + \beta = 6 , \alpha \beta = \color{#FF6800}{ 1 }$
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