1. **State the problem:** Solve the quadratic equation $x^2 + 20x + 82 = 7$ by completing the square. 2. **Rewrite the equation:** Move the constant term on the left side to the right side: $$x^2 + 20x = 7 - 82$$ $$x^2 + 20x = -75$$ 3. **Complete the square:** Take half of the coefficient of $x$, which is 20, divide by 2 to get 10, then square it to get $10^2 = 100$. 4. **Add 100 to both sides to complete the square:** $$x^2 + 20x + \cancel{100} = -75 + \cancel{100}$$ $$x^2 + 20x + 100 = 25$$ 5. **Rewrite the left side as a perfect square:** $$(x + 10)^2 = 25$$ 6. **Take the square root of both sides:** $$x + 10 = \pm \sqrt{25}$$ $$x + 10 = \pm 5$$ 7. **Solve for $x$:** - For the positive root: $$x + 10 = 5$$ $$x = 5 - 10 = -5$$ - For the negative root: $$x + 10 = -5$$ $$x = -5 - 10 = -15$$ 8. **Order the solutions from least to greatest:** $$x = -15, -5$$