Question

if x=√3+√2/√3-√2 and y =√3-√2/√3+√2, then find the value of x^2+y^2 -10xy

Answer 1

Steps to find the answer:

Step 1: Find x²

Step 2: Find y²

Step 3: Find 10xy

Step 4: Then we combine

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Step 1: Find x²

$x = \dfrac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}$

Rationalise the denominator:

$x = \dfrac{(\sqrt{3} + \sqrt{2}) (\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2}) (\sqrt{3} + \sqrt{2})}$

$x = (\sqrt{3} + \sqrt{2} )^2$

$x = (\sqrt{3})^2 + 2 \sqrt{3}\sqrt{2} + (\sqrt{2} )^2$

$x = 5 + 2\sqrt{6}$

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Step 2: Find y²

$y = \dfrac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}$

Rationalise the denominator:

$y = \dfrac{(\sqrt{3} - \sqrt{2}) (\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2}) (\sqrt{3} - \sqrt{2})}$

$y = (\sqrt{3} - \sqrt{2} )^2$

$y = 5 - 2\sqrt{6}$

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Step 3: Find 10xy:

$10xy = 10 (5 + 2\sqrt{6})(5 - 2\sqrt{6} )$

$10xy = 10 ( (5)^2 - (2\sqrt{6})^2 )$

$10xy = 10$

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Step 4: x² + y² - 10xy:

$x^2 + y^2 - 10xy = (5 + 2\sqrt{6} )^2 + (5 - 2\sqrt{6} )^2 - 10$

$x^2 + y^2 - 10xy = 25 + 2(5)(2\sqrt{6}) + (2\sqrt{6} )^2 + 25 - 2(5)(2\sqrt{6}) + (2\sqrt{6} )^2 - 10$

$x^2 + y^2 - 10xy = 25 + 24 + 25 + 24 - 10$

$x^2 + y^2 - 10xy = 88$

Answer: 88