# Calculate: (1)/(x)+(1)/(x+5)=(1)/(10)

## Expression: $\frac{ 1 }{ x }+\frac{ 1 }{ x+5 }=\frac{ 1 }{ 10 }$

Determine the defined range

$\begin{array} { l }\frac{ 1 }{ x }+\frac{ 1 }{ x+5 }=\frac{ 1 }{ 10 },& \begin{array} { l }x≠0,& x≠-5\end{array}\end{array}$

Move the constant to the left-hand side and change its sign

$\frac{ 1 }{ x }+\frac{ 1 }{ x+5 }-\frac{ 1 }{ 10 }=0$

Write all numerators above the least common denominator $10x \times \left( x+5 \right)$

$\frac{ 10\left( x+5 \right)+10x-x \times \left( x+5 \right) }{ 10x \times \left( x+5 \right) }=0$

Distribute $10$ through the parentheses

$\frac{ 10x+50+10x-x \times \left( x+5 \right) }{ 10x \times \left( x+5 \right) }=0$

Distribute $-x$ through the parentheses

$\frac{ 10x+50+10x-{x}^{2}-5x }{ 10x \times \left( x+5 \right) }=0$

Collect like terms

$\frac{ 15x+50-{x}^{2} }{ 10x \times \left( x+5 \right) }=0$

When the quotient of expressions equals $0$, the numerator has to be $0$

$15x+50-{x}^{2}=0$

Use the commutative property to reorder the terms

$-{x}^{2}+15x+50=0$

Change the signs on both sides of the equation

${x}^{2}-15x-50=0$

Identify the coefficients $a$, $b$ and $c$ of the quadratic equation

$\begin{array} { l }a=1,& b=-15,& c=-50\end{array}$

Substitute $a=1$, $b=-15$ and $c=-50$ into the quadratic formula $x=\frac{ -b\pm\sqrt{ {b}^{2}-4ac } }{ 2a }$

$x=\frac{ -\left( -15 \right)\pm\sqrt{ {\left( -15 \right)}^{2}-4 \times 1 \times \left( -50 \right) } }{ 2 \times 1 }$

Any expression multiplied by $1$ remains the same

$x=\frac{ -\left( -15 \right)\pm\sqrt{ {\left( -15 \right)}^{2}-4 \times \left( -50 \right) } }{ 2 \times 1 }$

Any expression multiplied by $1$ remains the same

$x=\frac{ -\left( -15 \right)\pm\sqrt{ {\left( -15 \right)}^{2}-4 \times \left( -50 \right) } }{ 2 }$

When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses

$x=\frac{ 15\pm\sqrt{ {\left( -15 \right)}^{2}-4 \times \left( -50 \right) } }{ 2 }$

Evaluate the power

$x=\frac{ 15\pm\sqrt{ 225-4 \times \left( -50 \right) } }{ 2 }$

Multiply the numbers

$x=\frac{ 15\pm\sqrt{ 225+200 } }{ 2 }$

$x=\frac{ 15\pm\sqrt{ 425 } }{ 2 }$

$x=\frac{ 15\pm5\sqrt{ 17 } }{ 2 }$

Write the solutions, one with a $+$ sign and one with a $-$ sign

$\begin{array} { l }\begin{array} { l }x=\frac{ 15+5\sqrt{ 17 } }{ 2 },\\x=\frac{ 15-5\sqrt{ 17 } }{ 2 }\end{array},& \begin{array} { l }x≠0,& x≠-5\end{array}\end{array}$

Check if the solution is in the defined range

$\begin{array} { l }x=\frac{ 15+5\sqrt{ 17 } }{ 2 },\\x=\frac{ 15-5\sqrt{ 17 } }{ 2 }\end{array}$

The equation has $2$ solutions

\begin{align*}&\begin{array} { l }x_1=\frac{ 15-5\sqrt{ 17 } }{ 2 },& x_2=\frac{ 15+5\sqrt{ 17 } }{ 2 }\end{array} \\&\begin{array} { l }x_1\approx-2.80776,& x_2\approx17.80776\end{array}\end{align*}

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