Solve for: (1)/(x)+(1)/(x+3)=(1)/(4)

Expression: $\frac{ 1 }{ x }+\frac{ 1 }{ x+3 }=\frac{ 1 }{ 4 }$

Determine the defined range

$\begin{array} { l }\frac{ 1 }{ x }+\frac{ 1 }{ x+3 }=\frac{ 1 }{ 4 },& \begin{array} { l }x≠0,& x≠-3\end{array}\end{array}$

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

$\frac{ 1 }{ x }+\frac{ 1 }{ x+3 }-\frac{ 1 }{ 4 }=0$

Write all numerators above the least common denominator $4x \times \left( x+3 \right)$

$\frac{ 4\left( x+3 \right)+4x-x \times \left( x+3 \right) }{ 4x \times \left( x+3 \right) }=0$

Distribute $4$ through the parentheses

$\frac{ 4x+12+4x-x \times \left( x+3 \right) }{ 4x \times \left( x+3 \right) }=0$

Distribute $-x$ through the parentheses

$\frac{ 4x+12+4x-{x}^{2}-3x }{ 4x \times \left( x+3 \right) }=0$

Collect like terms

$\frac{ 5x+12-{x}^{2} }{ 4x \times \left( x+3 \right) }=0$

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

$5x+12-{x}^{2}=0$

Use the commutative property to reorder the terms

$-{x}^{2}+5x+12=0$

Change the signs on both sides of the equation

${x}^{2}-5x-12=0$

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

$\begin{array} { l }a=1,& b=-5,& c=-12\end{array}$

Substitute $a=1$, $b=-5$ and $c=-12$ into the quadratic formula $x=\frac{ -b\pm\sqrt{ {b}^{2}-4ac } }{ 2a }$

$x=\frac{ -\left( -5 \right)\pm\sqrt{ {\left( -5 \right)}^{2}-4 \times 1 \times \left( -12 \right) } }{ 2 \times 1 }$

Any expression multiplied by $1$ remains the same

$x=\frac{ -\left( -5 \right)\pm\sqrt{ {\left( -5 \right)}^{2}-4 \times \left( -12 \right) } }{ 2 \times 1 }$

Any expression multiplied by $1$ remains the same

$x=\frac{ -\left( -5 \right)\pm\sqrt{ {\left( -5 \right)}^{2}-4 \times \left( -12 \right) } }{ 2 }$

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

$x=\frac{ 5\pm\sqrt{ {\left( -5 \right)}^{2}-4 \times \left( -12 \right) } }{ 2 }$

Evaluate the power

$x=\frac{ 5\pm\sqrt{ 25-4 \times \left( -12 \right) } }{ 2 }$

Multiply the numbers

$x=\frac{ 5\pm\sqrt{ 25+48 } }{ 2 }$

Add the numbers

$x=\frac{ 5\pm\sqrt{ 73 } }{ 2 }$

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

$\begin{array} { l }\begin{array} { l }x=\frac{ 5+\sqrt{ 73 } }{ 2 },\\x=\frac{ 5-\sqrt{ 73 } }{ 2 }\end{array},& \begin{array} { l }x≠0,& x≠-3\end{array}\end{array}$

Check if the solution is in the defined range

$\begin{array} { l }x=\frac{ 5+\sqrt{ 73 } }{ 2 },\\x=\frac{ 5-\sqrt{ 73 } }{ 2 }\end{array}$

The equation has $2$ solutions

$\begin{align*}&\begin{array} { l }x_1=\frac{ 5-\sqrt{ 73 } }{ 2 },& x_2=\frac{ 5+\sqrt{ 73 } }{ 2 }\end{array} \\&\begin{array} { l }x_1\approx-1.772,& x_2\approx6.772\end{array}\end{align*}$