1. **State the problem:** Solve the equation $$x^2 - 4 = 15x (x - 1)$$ for $x$. 2. **Rewrite the equation:** Expand the right side: $$x^2 - 4 = 15x^2 - 15x$$ 3. **Bring all terms to one side:** $$x^2 - 4 - 15x^2 + 15x = 0$$ 4. **Combine like terms:** $$-14x^2 + 15x - 4 = 0$$ 5. **Multiply both sides by $-1$ to simplify:** $$14x^2 - 15x + 4 = 0$$ 6. **Use the quadratic formula:** For $ax^2 + bx + c = 0$, solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=14$, $b=-15$, $c=4$. 7. **Calculate the discriminant:** $$\Delta = (-15)^2 - 4 \times 14 \times 4 = 225 - 224 = 1$$ 8. **Find the roots:** $$x = \frac{15 \pm \sqrt{1}}{2 \times 14} = \frac{15 \pm 1}{28}$$ 9. **Evaluate each root:** - $$x_1 = \frac{15 + 1}{28} = \frac{16}{28} = \frac{4}{7}$$ - $$x_2 = \frac{15 - 1}{28} = \frac{14}{28} = \frac{1}{2}$$ **Final answer:** The solutions to the equation are $$x = \frac{4}{7}$$ and $$x = \frac{1}{2}$$.