1. **State the problem:** Given the quadratic equation $$px^2 - 4x - \frac{21}{p} = 0$$ where $p$ is a constant, find the solutions for $x$ in terms of $p$. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a = p$$, $$b = -4$$, and $$c = -\frac{21}{p}$$. 4. **Substitute into the quadratic formula:** $$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(p)\left(-\frac{21}{p}\right)}}{2p}$$ 5. **Simplify inside the square root:** $$x = \frac{4 \pm \sqrt{16 - 4p \times \left(-\frac{21}{p}\right)}}{2p}$$ 6. **Cancel terms inside the root:** Since $p$ cancels: $$x = \frac{4 \pm \sqrt{16 + 84}}{2p}$$ 7. **Calculate the discriminant:** $$x = \frac{4 \pm \sqrt{100}}{2p}$$ 8. **Evaluate the square root:** $$x = \frac{4 \pm 10}{2p}$$ 9. **Find the two solutions:** - For the plus sign: $$x = \frac{4 + 10}{2p} = \frac{14}{2p} = \frac{7}{p}$$ - For the minus sign: $$x = \frac{4 - 10}{2p} = \frac{-6}{2p} = -\frac{3}{p}$$ **Final answer:** $$x = \frac{7}{p} \quad \text{or} \quad x = -\frac{3}{p}$$