the strategies listed, which is most appropriate for solving \( q^{2}-10 q=20 \) ? square root property zero product property What are the square Simplify the radical. Enter your answers in the boxes. The solutions are \( \square \pm \square . ~ \)

To solve the quadratic equation \( q^{2} - 10q = 20 \), let's follow the appropriate steps and fill in the solutions accordingly. ### 1. **Choose the Appropriate Strategy** Given the equation \( q^{2} - 10q = 20 \), the **best strategy** to solve this is **completing the square**, which is closely related to the **square root property**. This method is more suitable here than the zero product property because the equation isn't initially factored. ### 2. **Solve the Equation by Completing the Square** Let's solve the equation step by step: 1. **Start with the original equation:** \[ q^2 - 10q = 20 \] 2. **Move the constant term to the other side:** \[ q^2 - 10q - 20 = 0 \quad \text{(Alternatively, keep the equation as } q^2 - 10q = 20 \text{)} \] 3. **Complete the square:** - Take half of the coefficient of \( q \): \( \frac{-10}{2} = -5 \). - Square it: \( (-5)^2 = 25 \). - Add 25 to both sides to complete the square: \[ q^2 - 10q + 25 = 20 + 25 \] \[ (q - 5)^2 = 45 \] 4. **Apply the square root property:** \[ q - 5 = \pm \sqrt{45} \] 5. **Simplify the radical:** \[ \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \] 6. **Solve for \( q \):** \[ q = 5 \pm 3\sqrt{5} \] ### 3. **Enter the Solutions in the Boxes** The solutions can be expressed as: \[ \boxed{5 \pm 3\sqrt{5}} \] So, filling in the blanks: - First box: **5** - Second box: **3√5** **Final Answer:** \[ q = \boxed{5} \pm \boxed{3\sqrt{5}} \]