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Formula
Number of solution
Relationship between roots and coefficients
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 
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ 7 } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 8 } \end{array}$
$\begin{array} {l} x = 7 \\ x = - 8 \end{array}$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = 113$
 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 
$\alpha + \beta = - 1 , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ 56 }$
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