Solve the quadratic equation $36x^2 = 121$ by applying the factoring method.

Step 1 — Write in standard form. Subtract $121$ from both sides: $36x^2 - 121 = 0$

Step 2 — Factor the quadratic expression. The binomial is a difference of squares. $(6x - 11)(6x + 11) = 0$

Step 3 — Apply the Zero Product Property. Set each factor equal to zero: $6x - 11 = 0 \quad \text{or} \quad 6x + 11 = 0$

Step 4 — Solve each linear equation: $x = \frac{11}{6} \quad \text{or} \quad x = -\frac{11}{6}$

Step 5 — Check both solutions by substituting into the original equation $36x^2 = 121$: For $x = \frac{11}{6}$: $36\left(\frac{11}{6}\right)^2 = 36\left(\frac{121}{36}\right) = 121$ ✓ For $x = -\frac{11}{6}$: $36\left(-\frac{11}{6}\right)^2 = 36\left(\frac{121}{36}\right) = 121$ ✓ The solutions are $x = \frac{11}{6}$ and $x = -\frac{11}{6}$.