Solve without using quadratic formula: $\frac{4}{3x+3} = \frac{12}{x^2 - 1}$.

Question

Solve without using quadratic formula: $\frac{4}{3x+3} = \frac{12}{x^2 - 1}$. Is there a way to solve this without using the quadratic formula? The quadratic formula is one of my biggest weaknesses, and I would gladly appreciate an easier way to solve this equation.

Answer 1

In general, you'll need to use the quadratic formula (or something equivalent like completing the square). If a problem forbids the use of a general technique like this, it's a sign to look for some additional structure.

In this problem, it's that the denominators have a factor in common: $$ 3x+3 = 3(x+1) \quad \textrm{and} \quad x^2-1=(x-1)(x+1) $$

Multiplying both sides of the equation by $(x+1)$ gives $ \frac{4}{3} = \frac{12}{x-1} $, which is solvable through the usual means.

EDIT: in response to "the quadratic formula is one of my biggest weaknesses". Avoiding it only makes it more of a weakness; the best way to become more comfortable with a concept is to use it more often.

Answer 2

You are supposed to factor the difference of squares in the denominator. Then the problem is easy.

Answer 3

Perhaps something like this: \begin{eqnarray*} \frac{4}{3x+3} & = & \frac{12}{x^{2}-1}\\ \frac{4}{3\left(x+1\right)} & = & \frac{12}{\left(x+1\right)\left(x-1\right)}\\ \frac{4}{3} & = & \frac{12}{\left(x-1\right)}\\ x & = & 10 \end{eqnarray*}