Math, asked by Geniuso, 1 month ago

If x = 2 + \sqrt{2} and y = 2 - \sqrt{2}, then find the value of (x² + y²), Please do not spam.

Answers

Answered by Fieldmarshal
5

Given:-

X=2+√2

Y=2-√2

To find:-

X²+Y²

Solution:-

First find out X² and Y² separately than add.

X²=(2+√2)²

X²=4+2+4√2

→6+4√2-----------------------------------(1)

Y²=(2-√2)²

→ 6-4√2-----------------------------------(2)

Now add (1) and (2)

X²+Y²=(6+4√2) +(6-4√2)

→ 12

Final answer

X²+Y²=12

Answered by Anonymous
37

Given :-

 \rm \: x = 2 +  \sqrt{2}

 \rm \: y = 2 -  \sqrt{2}

To find :-

 \rm \: x {}^{2}  + y {}^{2}

Solution :-

 \longrightarrow \rm \: (2 +  \sqrt{2} ) {}^{2}  + (2 -  \sqrt{2} ) {}^{2}

Expanding the terms by using (a + b)² = a² + 2ab + b²

(a-b)² = a² -2ab + b²

 \implies \rm(2 +  \sqrt{2} ) {}^{2}

 \implies \:  \rm \: (2 ) {}^{2} +  (\sqrt{2} ) {}^{2}  + 2(2)( \sqrt{2} )

 \implies \:  \rm \: 4 +  2 + 4\sqrt{2}

 \implies \:  \rm \: 6+ 4\sqrt{2}

 \boxed{ \rm \: x {}^{2}  = (2 +  \sqrt{2} ) {}^{2}  = 6 + 4 \sqrt{2} }

 \implies \rm(2  -  \sqrt{2} ) {}^{2}

 \implies \:  \rm \: (2 ) {}^{2} +  (\sqrt{2} ) {}^{2}   -  2(2)( \sqrt{2} )

 \implies \:  \rm \: 4 +  2  -  4\sqrt{2}

 \implies \:  \rm \: 6 - 4\sqrt{2}

 \boxed{ \rm \: y {}^{2} = (2  -   \sqrt{2} ) {}^{2}  = 6  -  4 \sqrt{2} }

So,

x² + y² =

 \rm \: 6 + 4 \sqrt{2}  + 6 - 4 \sqrt{2}

  \rm \:  = 12

So,

x² + y² = 12

Know more Algebraic Identities:-

  • ( a + b)² = a² + 2ab + b²
  • ( a- b)² = a² -2ab + b²
  • ( a+ b)(a- b) = a² -b²
  • (a + b)² + (a -b)² = 2(a² + b²)
  • ( a + b)² - ( a- b)² = 4ab
  • ( a + b)³ = a³ + 3ab( a +b) + b³
  • ( a- b)³ = a³ -3ab( a-b) - b³
  • If a + b + c = 0 then a³ + b³ + c³ = 3abc
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