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Componendo and Dividendo

If x= √a+3b+√...

Question

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

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Solution

$S i n c e space x over 1 equals fraction numerator square root of a plus 3 b end root plus square root of a minus 3 b end root over denominator square root of a plus 3 b end root minus square root of a minus 3 b end root end fraction A p l l y i n g space c o m p o n e n d o space a n d space d i v i d e n d o space w e space g e t fraction numerator x plus 1 over denominator x minus 1 end fraction equals fraction numerator square root of a plus 3 b end root plus square root of a minus 3 b end root plus square root of a plus 3 b end root minus square root of a minus 3 b end root over denominator square root of a plus 3 b end root plus square root of a minus 3 b end root minus square root of a plus 3 b end root plus square root of a minus 3 b end root end fraction fraction numerator x plus 1 over denominator x minus 1 end fraction equals fraction numerator 2 square root of a plus 3 b end root over denominator 2 square root of a minus 3 b end root end fraction s q u a r i n g space b o t h space s i d e s fraction numerator x squared plus 2 x plus 1 over denominator x squared minus 2 x plus 1 end fraction equals fraction numerator a plus 3 b over denominator a minus 3 b end fraction A g a i n space a p p l y i n g space c o m p o n e n d o space a n d space d i v i d e n d o comma fraction numerator x squared plus 2 x plus 1 plus x squared minus 2 x plus 1 over denominator x squared plus 2 x plus 1 minus x squared plus 2 x minus 1 end fraction equals fraction numerator a plus 3 b plus a minus 3 b over denominator a plus 3 b minus a plus 3 b end fraction fraction numerator 2 left parenthesis x squared plus 1 right parenthesis over denominator 2 left parenthesis 2 x right parenthesis end fraction equals fraction numerator 2 left parenthesis a right parenthesis over denominator 2 left parenthesis 3 b right parenthesis end fraction 3 b left parenthesis x squared plus 1 right parenthesis equals 2 a x 3 b x squared plus 3 b equals 2 a x 3 b x squared minus 2 a x plus 3 b equals 0$

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x = \frac{\sqrt{a + 3 b} + \sqrt{a - 3 b}}{\sqrt{a + 3 b} - \sqrt{a - 3 b}}

3 b x^{2} - 2 a x + 3 b = 0

\frac{x}{1} = \frac{\sqrt{a + 3 b} + \sqrt{a - 3 b}}{\sqrt{a + 3 b} - \sqrt{a - 3 b}}

, then prove that

3 b x^{2} + 3 b - 2 a x = 0

.

\frac{x}{1} = \frac{\sqrt{a + 3 b} + \sqrt{a - 3 b}}{\sqrt{a + 3 b} - \sqrt{a - 3 b}}

, then, show that

3 b x^{2} + 3 b - 2 a x = 0

.

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

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

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