1. **State the problem:** Solve the equation $$45(p^2 - 1) = 56p$$ for $p$. 2. **Expand and rearrange the equation:** $$45p^2 - 45 = 56p$$ 3. **Bring all terms to one side to set the equation to zero:** $$45p^2 - 56p - 45 = 0$$ 4. **Use the quadratic formula:** The quadratic formula for $ax^2 + bx + c = 0$ is $$p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=45$, $b=-56$, and $c=-45$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-56)^2 - 4 \times 45 \times (-45) = 3136 + 8100 = 11236$$ 6. **Calculate the square root of the discriminant:** $$\sqrt{11236} = 106$$ 7. **Substitute values into the quadratic formula:** $$p = \frac{-(-56) \pm 106}{2 \times 45} = \frac{56 \pm 106}{90}$$ 8. **Find the two solutions:** - For the plus sign: $$p = \frac{56 + 106}{90} = \frac{162}{90} = \frac{\cancel{162}}{\cancel{90}} = \frac{9}{5} = 1.8$$ - For the minus sign: $$p = \frac{56 - 106}{90} = \frac{-50}{90} = \frac{\cancel{-50}}{\cancel{90}} = \frac{-5}{9} \approx -0.5556$$ **Final answer:** $$p = \frac{9}{5} \quad \text{or} \quad p = -\frac{5}{9}$$