1. **State the problem:** Solve the equation $$16(5 - 2x)^2 + 33 = 42$$ for $x$. 2. **Isolate the squared term:** Subtract 33 from both sides: $$16(5 - 2x)^2 + 33 - 33 = 42 - 33$$ $$16(5 - 2x)^2 = 9$$ 3. **Divide both sides by 16:** $$\cancel{16}(5 - 2x)^2 = \frac{9}{\cancel{16}}$$ $$ (5 - 2x)^2 = \frac{9}{16}$$ 4. **Take the square root of both sides:** Remember to consider both positive and negative roots. $$5 - 2x = \pm \frac{3}{4}$$ 5. **Solve for $x$ in each case:** - Case 1: $$5 - 2x = \frac{3}{4}$$ $$-2x = \frac{3}{4} - 5 = \frac{3}{4} - \frac{20}{4} = -\frac{17}{4}$$ $$x = \frac{-\frac{17}{4}}{-2} = \frac{17}{8}$$ - Case 2: $$5 - 2x = -\frac{3}{4}$$ $$-2x = -\frac{3}{4} - 5 = -\frac{3}{4} - \frac{20}{4} = -\frac{23}{4}$$ $$x = \frac{-\frac{23}{4}}{-2} = \frac{23}{8}$$ 6. **Final answer:** $$x = \frac{17}{8} \quad \text{or} \quad x = \frac{23}{8}$$