If

$$\sqrt{5x + 9} + \sqrt{5x - 9} = 3(2 + \sqrt{2}),$$

then

$$\sqrt{10x + 9} ext{ is equal to }$$

Question

If

$$\sqrt{5x + 9} + \sqrt{5x - 9} = 3(2 + \sqrt{2}),$$

then

$$\sqrt{10x + 9} 	ext{ is equal to }$$

Accepted Answer

**Solution:**

1. Given:
   $$\sqrt{5x + 9} + \sqrt{5x - 9} = 3(2 + \sqrt{2}) = 6 + 3\sqrt{2}$$

2. Express the right-hand side (RHS) as sum of square roots:
   - Note that $6 = \sqrt{36}$ and $3\sqrt{2} = \sqrt{18}$ because $3\sqrt{2} = \sqrt{9 	imes 2} = \sqrt{18}$.
   - So, RHS can be written as:
     $$\sqrt{36} + \sqrt{18}$$

3. Equate the terms under the square roots on the left-hand side (LHS) and RHS:
   - Since the LHS is $\sqrt{5x + 9} + \sqrt{5x - 9}$ and RHS is $\sqrt{36} + \sqrt{18}$, we can match:
     $$5x + 9 = 36$$
     $$5x - 9 = 18$$

4. Solve for $x$ using the first equation:
   $$5x + 9 = 36 \implies 5x = 27 \implies x = \frac{27}{5} = 5.4$$

5. Verify with the second equation:
   $$5x - 9 = 18 \implies 5x = 27 \implies x = 5.4$$
   - Both equations give the same $x$, so $x = 5.4$ is correct.

6. Calculate $\sqrt{10x + 9}$:
   $$10x + 9 = 10 	imes 5.4 + 9 = 54 + 9 = 63$$

7. Simplify $\sqrt{63}$:
   $$\sqrt{63} = \sqrt{9 	imes 7} = 3\sqrt{7}$$

**Answer:** $3\sqrt{7}$

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