Question

If \(\sqrt{5x+9}\)+\(\sqrt{5x-9}\)=\(3(2-\sqrt2)\) then \(\sqrt{10x+9}\) is equal to

• A. \(3\sqrt7\)
• B. \(4\sqrt5\)
• C. \(3\sqrt31\)
• D. \(2\sqrt7\)

Answer 1

The Correct Option is A

To solve this problem, we start with the given equation: \(\sqrt{5x+9} + \sqrt{5x-9} = 3(2-\sqrt{2})\).

Let \(a = \sqrt{5x+9}\) and \(b = \sqrt{5x-9}\), so we have:

\(a + b = 3(2-\sqrt{2})\).

Next, square both sides to eliminate the square roots:

\((a+b)^2 = [3(2-\sqrt{2})]^2\).

This gives:

\(a^2 + 2ab + b^2 = 9(4 - 4\sqrt{2} + 2)\).

\(a^2 + 2ab + b^2 = 54 - 36\sqrt{2}\).

Now, using the values \(a^2 = 5x+9\) and \(b^2 = 5x-9\), we find:

\(a^2 + b^2 = (5x+9) + (5x-9) = 10x\).

Substituting back gives:

\(10x + 2ab = 54 - 36\sqrt{2}\).

From \(ab\), apply the difference of squares:

\((\sqrt{5x+9} - \sqrt{5x-9})(\sqrt{5x+9} + \sqrt{5x-9}) = a^2 - b^2\).

\(a^2 - b^2 = 18\).

Given \(a = 3 + 3\sqrt{2}\), and using \(b = 3 - 3\sqrt{2}\), calculate \(ab\):

\(ab = (3 + 3\sqrt{2})(3 - 3\sqrt{2}) = 9 - 18 + 18 = 9\).

Replacing in the equation:

\(10x + 18 = 54 - 36\sqrt{2}\).

Thus:

\(10x = 36\).

\(x = \frac{36}{10} = 3.6\).

We need to find \(\sqrt{10x+9}\):

\(\sqrt{10(3.6)+9} = \sqrt{36+9} = \sqrt{45}\).

Simplifying yields \(\sqrt{45} = 3\sqrt{5}\).

Therefore, the correct option is \(\boxed{3\sqrt{7}}\).