$\begin{array} { l }\frac{ 1 }{ x }+\frac{ 1 }{ x+5 }=\frac{ 1 }{ 10 },& \begin{array} { l }x≠0,& x≠-5\end{array}\end{array}$
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$x=\frac{ 15\pm\sqrt{ 225+200 } }{ 2 }$
Add the numbers$x=\frac{ 15\pm\sqrt{ 425 } }{ 2 }$
Simplify the radical expression$x=\frac{ 15\pm5\sqrt{ 17 } }{ 2 }$
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$\begin{align*}&\begin{array} { l }x_1=\frac{ 15-5\sqrt{ 17 } }{ 2 },& x_2=\frac{ 15+5\sqrt{ 17 } }{ 2 }\end{array} \\&\begin{array} { l }x_1\approx-2.80776,& x_2\approx17.80776\end{array}\end{align*}$