Question

x=√2+1/√2-1
y=√2-1/√2+1
find x²+y²+xy​

Answer 1

x²+y²+xy= 35

Step-by-step explanation:

Given Information -

• ${x=\frac{√2+1}{√2-1}}$

• ${y=\frac{√2-1}{√2+1}}$

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To Find-

• x²+y²+xy

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Solution -

Step-1= Rationalising 'x' -

$x = \frac{ \sqrt{2} + 1}{ \sqrt{2} - 1 } \\ \\ = \frac{ \sqrt{2} + 1}{ \sqrt{2} - 1} \times \frac{ \sqrt{2} + 1 }{ \sqrt{2} + 1 } \\ \\ = \frac{( \sqrt{2} + 1 {)}^{2} }{( \sqrt{2} + 1)( \sqrt{2} - 1) } \\ \\ = \frac{( \sqrt{2} {)}^{2} + (1 {)}^{2} + 2(1 \sqrt{2} ) }{( \sqrt{2} - 1 {)}^{2} } \\ \\ = \frac{2 + 1 + 2 \sqrt{2} }{1} \\ \\ = 3 + 2 \sqrt{2}$

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Step-2= Rationalising 'y' -

$y = \frac{ \sqrt{2} - 1 }{ \sqrt{2} + 1 } \\ \\ = \frac{ \sqrt{2} - 1 }{ \sqrt{2} + 1} \times \frac{ \sqrt{2} - 1 }{ \sqrt{2} - 1} \\ \\ = \frac{( \sqrt{2} - 1 {)}^{2} }{( \sqrt{2} + 1)( \sqrt{2} - 1)} \\ \\ = \frac{( \sqrt{2} {)}^{2} + (1 {)}^{2} - 2(1 \sqrt{2} ) }{( \sqrt{2} {)}^{2} - (1 {)}^{2} } \\ \\ = \frac{3 - 2 \sqrt{2} }{1} \\ \\ = 3 - 2 \sqrt{2}$

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Step-3(Final Step)= Solving x²+y²+xy-

${x}^{2} + {y}^{2} + xy \\ \\ = (x + y {)}^{2} - xy \\ \\ = (3 + 2 \sqrt{2} + 3 - 2\sqrt{2} {)}^{2} - ((3 + 2\sqrt{2} )(3 - 2 \sqrt{2} )) \\ \\ = (6 {)}^{2} - ((3 - 2 \sqrt{2} {)}^{2} ))\\ \\ = 36 - ((3 {)}^{2} - (2 \sqrt{2} {)}^{2} ) \\ \\ = 36 - (9 - 8) \\ \\ = 36 - 1 \\ \\ = 35$