Solve the quadratic equation $169x^2 = 49$ by applying the factoring method.

Step 1 — Write in standard form. Subtract $49$ from both sides: $169x^2 - 49 = 0$

Step 2 — Factor the quadratic expression. The binomial is a difference of squares. $(13x - 7)(13x + 7) = 0$

Step 3 — Apply the Zero Product Property. Set each factor equal to zero: $13x - 7 = 0 \quad \text{or} \quad 13x + 7 = 0$

Step 4 — Solve each linear equation: $x = \frac{7}{13} \quad \text{or} \quad x = -\frac{7}{13}$

Step 5 — Check both solutions by substituting into the original equation $169x^2 = 49$: For $x = \frac{7}{13}$: $169\left(\frac{7}{13}\right)^2 = 169\left(\frac{49}{169}\right) = 49$ ✓ For $x = -\frac{7}{13}$: $169\left(-\frac{7}{13}\right)^2 = 169\left(\frac{49}{169}\right) = 49$ ✓ The solutions are $x = \frac{7}{13}$ and $x = -\frac{7}{13}$.