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 $$ \square \pm \square . ~ $$

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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:**
$$
q = \boxed{5} \pm \boxed{3\sqrt{5}}
$$
The solutions are $$ 5 \pm 3\sqrt{5} $$.

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 \( \square \pm \square . ~ \)

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