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

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

Find: $\sqrt{10x + 9}$

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First, let's expand the right side:

$3(2+\sqrt{2}) = 6 + 3\sqrt{2}$

So our equation becomes:

$\sqrt{5x+9}+\sqrt{5x-9} = 6 + 3\sqrt{2}$

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Here's the crucial step that makes this problem elegant. We need to recognize that $6 + 3\sqrt{2}$ can be written as the sum of two square roots.

Let's check: Can we write $6 + 3\sqrt{2} = \sqrt{a} + \sqrt{b}$ for some integers $a$ and $b$?

Let's try $\sqrt{36} + \sqrt{18}$:

$\sqrt{36} = 6$

$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

Perfect! So: $6 + 3\sqrt{2} = \sqrt{36} + \sqrt{18}$

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Now our equation becomes:

$\sqrt{5x+9}+\sqrt{5x-9} = \sqrt{36} + \sqrt{18}$

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This is where students often get confused. Why can we match terms directly?

The reasoning: Notice the structure on both sides:

Left side: $\sqrt{5x+9}$ and $\sqrt{5x-9}$ (difference of 18 inside)

Right side: $\sqrt{36}$ and $\sqrt{18}$ (difference of 18 inside)

When we have expressions with the same structure, we can match corresponding terms:

$\sqrt{5x+9} = \sqrt{36}$ (the larger terms)

$\sqrt{5x-9} = \sqrt{18}$ (the smaller terms)

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From $\sqrt{5x+9} = \sqrt{36}$:

$5x + 9 = 36$

$5x = 27$

$x = \dfrac{27}{5}$

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Now we can find $\sqrt{10x + 9}$:

$\sqrt{10x + 9} = \sqrt{10 \times \dfrac{27}{5} + 9}$

$= \sqrt{\dfrac{270}{5} + 9}$

$= \sqrt{54 + 9}$

$= \sqrt{63}$

$= \sqrt{9 \times 7}$

$= 3\sqrt{7}$

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Therefore: $\sqrt{10x + 9} = 3\sqrt{7}$

Key Takeaway for Future Problems:

When you see equations involving sums of square roots, try to express both sides in similar forms. This pattern recognition technique often transforms complex radical equations into simple algebraic ones!