A Positive root of `32 + 4 sqrt15` is

Options

A. `sqrt30 + sqrt2`
B. `sqrt30`
C. `sqrt2`
D. cannot determine

Answer

To find the positive root of the expression \( 32 + 4\sqrt{15} \), we can express it in the form of a perfect square. Let's go through the steps to solve this.

### Step-by-Step Solution:

1. **Assume the Expression is a Perfect Square:**
We assume that \( 32 + 4\sqrt{15} \) can be expressed as \( (a + b)^2 \).

2. **Expand the Perfect Square:**
Expanding \( (a + b)^2 \) gives us:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
We need to match this with \( 32 + 4\sqrt{15} \).

3. **Set Up the Equations:**
From the expansion, we can set up the following equations:
\[ a^2 + b^2 = 32 \quad \text{(1)} \]
\[ 2ab = 4\sqrt{15} \quad \text{(2)} \]

4. **Simplify Equation (2):**
From equation (2), we can simplify to find \( ab \):
\[ ab = 2\sqrt{15} \quad \text{(3)} \]

5. **Substitute \( b \) in Terms of \( a \):**
From equation (3), we can express \( b \) in terms of \( a \):
\[ b = \frac{2\sqrt{15}}{a} \quad \text{(4)} \]

6. **Substitute \( b \) into Equation (1):**
Substitute equation (4) into equation (1):
\[ a^2 + \left(\frac{2\sqrt{15}}{a}\right)^2 = 32 \]
This simplifies to:
\[ a^2 + \frac{4 \cdot 15}{a^2} = 32 \]
\[ a^2 + \frac{60}{a^2} = 32 \]

7. **Multiply through by \( a^2 \):**
To eliminate the fraction, multiply through by \( a^2 \):
\[ a^4 - 32a^2 + 60 = 0 \]

8. **Let \( x = a^2 \):**
Let \( x = a^2 \), then the equation becomes:
\[ x^2 - 32x + 60 = 0 \]

9. **Solve the Quadratic Equation:**
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{32 \pm \sqrt{(-32)^2 - 4 \cdot 1 \cdot 60}}{2 \cdot 1} \]
\[ x = \frac{32 \pm \sqrt{1024 - 240}}{2} \]
\[ x = \frac{32 \pm \sqrt{784}}{2} \]
\[ x = \frac{32 \pm 28}{2} \]
This gives us two possible values for \( x \):
\[ x = \frac{60}{2} = 30 \quad \text{or} \quad x = \frac{4}{2} = 2 \]

10. **Find \( a \) and \( b \):**
Since \( x = a^2 \):
- If \( x = 30 \), then \( a = \sqrt{30} \).
- If \( x = 2 \), then \( a = \sqrt{2} \).

11. **Calculate \( b \):**
Using equation (4) to find \( b \):
- For \( a = \sqrt{30} \):
\[ b = \frac{2\sqrt{15}}{\sqrt{30}} = \sqrt{2} \]
- For \( a = \sqrt{2} \):
\[ b = \frac{2\sqrt{15}}{\sqrt{2}} = \sqrt{30} \]

12. **Conclusion:**
The positive root of \( 32 + 4\sqrt{15} \) can be expressed as:
\[ a + b = \sqrt{30} + \sqrt{2} \]