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Formula
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
$y = x ^ { 2 } + \left ( x + 1 \right ) ^ { 2 }$
$y = 113$
$y$-intercept
$\left ( 0 , 1 \right )$
Minimum
$\left ( - \dfrac { 1 } { 2 } , \dfrac { 1 } { 2 } \right )$
Standard form
$y = 2 \left ( x + \dfrac { 1 } { 2 } \right ) ^ { 2 } + \dfrac { 1 } { 2 }$
$x ^{ 2 } + \left( x+1 \right) ^{ 2 } = 113$
$\begin{array} {l} x = 7 \\ x = - 8 \end{array}$
Find solution by method of factorization
$x ^ { 2 } + \left ( x + 1 \right ) ^ { 2 } = \color{#FF6800}{ 113 }$
$ $ Move the expression to the left side and change the symbol $ $
$x ^ { 2 } + \left ( x + 1 \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 113 } = 0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 113 } = 0$
$ $ Expand the expression $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 112 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 112 } = 0$
$acx^{2} + \left(ad + bc\right)x +bd = \left(ax + b\right)\left(cx+d\right)$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } \right ) \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \right ) = 0$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } \right ) \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 8 } \right ) = \color{#FF6800}{ 0 }$
$ $ If the product of the factor is 0, at least one factor should be 0 $ $
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 8 } = \color{#FF6800}{ 0 } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 7 } = \color{#FF6800}{ 0 } \\ \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 8 } = \color{#FF6800}{ 0 } \end{array}$
$ $ Solve the equation to find $ x$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 7 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 8 } \end{array}$
$\begin{array} {l} x = 7 \\ x = - 8 \end{array}$
Solve quadratic equations using the square root
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = 113$
$ $ Organize the expression $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = 113$
$2 x ^ { 2 } + 2 x + 1 = \color{#FF6800}{ 113 }$
$ $ Move the expression to the left side and change the symbol $ $
$2 x ^ { 2 } + 2 x + 1 \color{#FF6800}{ - } \color{#FF6800}{ 113 } = 0$
$2 x ^ { 2 } + 2 x + \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 113 } = 0$
$ $ Subtract $ 113 $ from $ 1$
$2 x ^ { 2 } + 2 x \color{#FF6800}{ - } \color{#FF6800}{ 112 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 112 } = \color{#FF6800}{ 0 }$
$ $ Divide both sides by the coefficient of the leading highest term $ $
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 56 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 56 } = \color{#FF6800}{ 0 }$
$ $ Convert the quadratic expression on the left side to a perfect square format $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 56 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 1 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$
$\left ( x + \dfrac { 1 } { 2 } \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 56 } \color{#FF6800}{ - } \left ( \color{#FF6800}{ \dfrac { 1 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = 0$
$ $ Move the constant to the right side and change the sign $ $
$\left ( x + \dfrac { 1 } { 2 } \right ) ^ { 2 } = \color{#FF6800}{ 56 } \color{#FF6800}{ + } \left ( \color{#FF6800}{ \dfrac { 1 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } }$
$\left ( x + \dfrac { 1 } { 2 } \right ) ^ { 2 } = 56 + \left ( \color{#FF6800}{ \dfrac { 1 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } }$
$ $ When raising a fraction to the power, raise the numerator and denominator each to the power $ $
$\left ( x + \dfrac { 1 } { 2 } \right ) ^ { 2 } = 56 + \dfrac { \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } } { \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 56 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 ^ { 2 } } { 2 ^ { 2 } } }$
$ $ Organize the expression $ $
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 225 } { 4 } }$
$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 2 } } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ \dfrac { 225 } { 4 } }$
$ $ Solve quadratic equations using the square root $ $
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 2 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 225 } { 4 } } }$
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 1 } { 2 } } = \pm \sqrt{ \color{#FF6800}{ \dfrac { 225 } { 4 } } }$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { 15 } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 2 } }$
$\color{#FF6800}{ x } = \pm \color{#FF6800}{ \dfrac { 15 } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 2 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 15 } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 15 } { 2 } } \end{array}$
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 15 } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 15 } { 2 } } \end{array}$
$ $ Organize the expression $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 7 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 8 } \end{array}$
$\begin{array} {l} x = 7 \\ x = - 8 \end{array}$
Calculate using the quadratic formula
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = 113$
$ $ Organize the expression $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = 113$
$2 x ^ { 2 } + 2 x + 1 = \color{#FF6800}{ 113 }$
$ $ Move the expression to the left side and change the symbol $ $
$2 x ^ { 2 } + 2 x + 1 \color{#FF6800}{ - } \color{#FF6800}{ 113 } = 0$
$2 x ^ { 2 } + 2 x + \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 113 } = 0$
$ $ Subtract $ 113 $ from $ 1$
$2 x ^ { 2 } + 2 x \color{#FF6800}{ - } \color{#FF6800}{ 112 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 112 } = 0$
$ $ Bind the expressions with the common factor $ 2$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 56 } \right ) = 0$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 56 } \right ) = \color{#FF6800}{ 0 }$
$ $ Divide both sides by $ 2$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 56 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 \pm \sqrt{ 1 ^ { 2 } - 4 \times 1 \times \left ( - 56 \right ) } } { 2 \times 1 } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 \pm \sqrt{ 225 } } { 2 \times 1 } }$
$x = \dfrac { - 1 \pm \sqrt{ \color{#FF6800}{ 225 } } } { 2 \times 1 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { - 1 \pm \color{#FF6800}{ 15 } } { 2 \times 1 }$
$x = \dfrac { - 1 \pm 15 } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
$ $ Multiplying any number by 1 does not change the value $ $
$x = \dfrac { - 1 \pm 15 } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 \pm 15 } { 2 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 + 15 } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 - 15 } { 2 } } \end{array}$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 15 } } { 2 } \\ x = \dfrac { - 1 - 15 } { 2 } \end{array}$
$ $ Add $ - 1 $ and $ 15$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 14 } } { 2 } \\ x = \dfrac { - 1 - 15 } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 14 } { 2 } } \\ x = \dfrac { - 1 - 15 } { 2 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 7 } { 1 } } \\ x = \dfrac { - 1 - 15 } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 7 } { 1 } } \\ x = \dfrac { - 1 - 15 } { 2 } \end{array}$
$ $ Reduce the fraction to the lowest term $ $
$\begin{array} {l} x = \color{#FF6800}{ 7 } \\ x = \dfrac { - 1 - 15 } { 2 } \end{array}$
$\begin{array} {l} x = 7 \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 15 } } { 2 } \end{array}$
$ $ Find the sum of the negative numbers $ $
$\begin{array} {l} x = 7 \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 16 } } { 2 } \end{array}$
$\begin{array} {l} x = 7 \\ x = \color{#FF6800}{ \dfrac { - 16 } { 2 } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = 7 \\ x = \color{#FF6800}{ \dfrac { - 8 } { 1 } } \end{array}$
$\begin{array} {l} x = 7 \\ x = \dfrac { - 8 } { \color{#FF6800}{ 1 } } \end{array}$
$ $ If the denominator is 1, the denominator can be removed $ $
$\begin{array} {l} x = 7 \\ x = \color{#FF6800}{ - } \color{#FF6800}{ 8 } \end{array}$
$ $ 2 real roots $ $
Find the number of solutions
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = 113$
$ $ Organize the expression $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = 113$
$2 x ^ { 2 } + 2 x + 1 = \color{#FF6800}{ 113 }$
$ $ Move the expression to the left side and change the symbol $ $
$2 x ^ { 2 } + 2 x + 1 \color{#FF6800}{ - } \color{#FF6800}{ 113 } = 0$
$2 x ^ { 2 } + 2 x + \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 113 } = 0$
$ $ Subtract $ 113 $ from $ 1$
$2 x ^ { 2 } + 2 x \color{#FF6800}{ - } \color{#FF6800}{ 112 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 112 } = \color{#FF6800}{ 0 }$
$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$
$\color{#FF6800}{ D } = \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 112 } \right )$
$D = \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - 4 \times 2 \times \left ( - 112 \right )$
$ $ Calculate power $ $
$D = \color{#FF6800}{ 4 } - 4 \times 2 \times \left ( - 112 \right )$
$D = 4 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 112 } \right )$
$ $ Multiply the numbers $ $
$D = 4 + \color{#FF6800}{ 896 }$
$D = \color{#FF6800}{ 4 } \color{#FF6800}{ + } \color{#FF6800}{ 896 }$
$ $ Add $ 4 $ and $ 896$
$D = \color{#FF6800}{ 900 }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 900 }$
$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $
$ $ 2 real roots $ $
$\alpha + \beta = - 1 , \alpha \beta = - 56$
Find the sum and product of the two roots of the quadratic equation
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = 113$
$ $ Organize the expression $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = 113$
$2 x ^ { 2 } + 2 x + 1 = \color{#FF6800}{ 113 }$
$ $ Move the expression to the left side and change the symbol $ $
$2 x ^ { 2 } + 2 x + 1 \color{#FF6800}{ - } \color{#FF6800}{ 113 } = 0$
$2 x ^ { 2 } + 2 x + \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 113 } = 0$
$ $ Subtract $ 113 $ from $ 1$
$2 x ^ { 2 } + 2 x \color{#FF6800}{ - } \color{#FF6800}{ 112 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 112 } = \color{#FF6800}{ 0 }$
$ $ 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}$
$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 2 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 112 } { 2 } }$
$\alpha + \beta = - \color{#FF6800}{ \dfrac { 2 } { 2 } } , \alpha \beta = \dfrac { - 112 } { 2 }$
$ $ Reduce the fraction $ $
$\alpha + \beta = - \color{#FF6800}{ 1 } , \alpha \beta = \dfrac { - 112 } { 2 }$
$\alpha + \beta = - 1 , \alpha \beta = \color{#FF6800}{ \dfrac { - 112 } { 2 } }$
$ $ Reduce the fraction $ $
$\alpha + \beta = - 1 , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ 56 }$
$ $ 그래프 보기 $ $
Graph
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search-thumbnail-What is the inverse of $g\left(x\right)=\sqrt{x-1} $ $Og\left(x\right)=\sqrt{x+1} $ $+1$ $g^{'}\left(x\right)=5-$ $y_{9}$ $g\sqrt{ef\left(x\left(nght\right)} =\left(f_{tac\left(x+1\right)0sq\pi \left(x\right)}$ $Og^{'}\left(x\right)=x^{2}+1$ $○g\left(x\right)=x^{2}-1$ « Previous Ne
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