Answer
$\left\{-2,7\right\}$
Work Step by Step
Multiplying by the $LCD= x ,$ the given equation $
\dfrac{14}{x}=x-5
,$ is equivalent to \begin{align*}\require{cancel} x\left(\dfrac{14}{x}\right)&=(x-5)x \\\\ 14&=x^2-5x \\\\ 0&=x^2-5x-14 \\ x^2-5x-14&=0 .\end{align*} Using the factoring of trinomials, the factored form of the equation above is \begin{align*} (x-7)(x+2)&=0 .\end{align*} Equating each factor to zero (Zero Product Property) and solving for the variable, then \begin{array}{l|r} x-7=0 & x+2=0 \\ x=7 & x=-2 .\end{array} Checking the solutions in the given equation results to \begin{array}{l|r} \text{If }x=7: & \text{If }x=-2: \\\\ \dfrac{14}{7}\overset{?}=7-5 & \dfrac{14}{-2}\overset{?}=-2-5 \\\\ 2\overset{\checkmark}=2 & -7\overset{\checkmark}=-7 .\end{array} Since both solutions satisfy the given equation, then the solution set of the equation $ \dfrac{14}{x}=x-5 $ is $\left\{-2,7\right\}$.
