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Operations on Rational and Irrational Numbers

If x=√2a+3b...

Question

If $x = \frac{\sqrt{2 a + 3 b} + \sqrt{2 a - 3 b}}{\sqrt{2 a + 3 b} - \sqrt{2 a - 3 b}}$ , the prove that $3 b x^{2} - 4 a x + 3 b = 0$ .

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Solution

$x = \frac{\sqrt{2 a + 3 b} + \sqrt{2 a - 3 b}}{\sqrt{2 a + 3 b} - \sqrt{2 a - 3 b}}$
Rationalizing
$\Rightarrow x = \frac{\sqrt{2 a + 3 b} + \sqrt{2 a - 3 b}}{\sqrt{2 a + 3 b} - \sqrt{2 a - 3 b}} \times \frac{\sqrt{2 a + 3 b} + \sqrt{2 a - 3 b}}{\sqrt{2 a + 3 b} + \sqrt{2 a - 3 b}}$
$\Rightarrow x = \frac{2 a + 3 b + 2 a - 3 b - 2 \sqrt{4 a^{2} - 9 b^{2}}}{2 a + 3 b - 2 a + 3 b}$
$\Rightarrow x = \frac{2 a - \sqrt{4 a^{2} - 9 b^{2}}}{36}$
$\Rightarrow 36 x - 2 a = - \sqrt{4 a^{2} - 9 b^{2}}$
squaring
$\Rightarrow 9 b^{2} x^{2} + 4 a^{2} - 12 a b c = 4 a^{2} - 9 b^{2}$
$\Rightarrow 96^{2} x^{2} + 96^{2} - 12 a b x = 0$
divide by $3 b$
$\Rightarrow 36 x^{2} - 4 a x + 36 = 0$

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