Quadratic Equations Question 7

$$ \frac{x+1}{2}=\frac{5}{x-2} $$

What is the solution set to the equation shown above?

{ -1, 2 }

{ -3, 4 }

{ -4, 3 }

{ -3, -1, 2, 4 }

Approach 1

We can cross multiply first to remove the denominators.

$$ \frac{x+1}{2}=\frac{5}{x-2} $$ $$ (x+1)(x-2)=5(2) $$ $$ x^2-x-2=10 $$ $$ x^2-x-12=0$$

Factoring, quadratic equation, and completing the square are all viable methods of obtaining the solution. Below is the factored form:

$$ (x-4)(x+3)=0$$ $$ x=4 \text{ and} -3 $$

Neither of these solutions are excluded in the domain of the initial equation and are therefore valid solutions.

Approach 2

We can test the solutions in the answer choices. For example, we can exclude a few options since $x=-1$ does not satisfy the equation.

$$ \frac{x+1}{2}=\frac{5}{x-2} $$ $$ \frac{-1+1}{2}=\frac{5}{-1-2} $$ $$ 0 \neq -\frac{5}{3} $$

Testing other solutions would lead us to the correct answer choice.