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If x = (√3-√2)/(√3+√2) and y = (√3+√2)/(√3-√2),
Find the value of x² + y² + xy.​

If X 3232 And Y 3232Find The Value Of X Y Xy class=

Sagot :

SalmonWorldTopper

Step-by-step explanation:

Given :-

x = (√3-√2)/(√3+√2) and

y = (√3+√2)/(√3-√2)

To find :-

Find the value of x²+y²+xy ?

Solution:-

Given that

x = (√3-√2)/(√3+√2)

The denominator = √3+√2

The Rationalising factor of √3+√2 is √3-√2

On Rationalising the denominator then

=>x=[(√3-√2)/(√3+√2)]×[(√3-√2)/(√3-√2)]

=> x = [(√3-√2)(√3-√2)]×[(√3+√2)(√3-√2)]

=> x = (√3-√2)²/[(√3+√2)(√3-√2)]

=> x = (√3-√2)²/[(√3)²-(√2)²]

Since (a+b)(a-b) = a²-b²

Where , a = √3 and b = √2

=> x = (√3-√2)²/(3-2)

=> x = (√3-√2)²/1

=> x = (√3-√2)²

=> x = (√3)²-2(√3)(√2)+(√2)²

Since , (a-b)² = a²-2ab+b²

Where , a = √3 and b = √2

=> x = 3-2√6+2

=> x = 5-2√6 --------------------(1)

On squaring both sides then

=> x² = (5-2√6)²

=> x² = 5²-2(5)(2√6)+(2√6)²

Since ,(a-b)² = a²-2ab+b²

Where , a = 5 and b = 2√6

=> x² = 25-20√6+24

=> x² = 49-20√6 ---------------(2)

and

y = (√3+√2)/(√3-√2)

The denominator = √3-√2

The Rationalising factor of √3-√2 is √3+√2

On Rationalising the denominator then

=>y=[(√3+√2)/(√3-√2)]×[(√3+√2)/(√3+√2)]

=>y=[(√3+√2)(√3+√2)]×[(√3-√2)(√3+√2)]

=> y = (√3+√2)²/[(√3-√2)(√3+√2)]

=> y = (√3+√2)²/[(√3)²-(√2)²]

Since (a+b)(a-b) = a²-b²

Where , a = √3 and b = √2

=> y = (√3+√2)²/(3-2)

=> y = (√3-√2)²/1

=> y = (√3+√2)²

=> y = (√3)²+2(√3)(√2)+(√2)²

Since , (a+b)² = a²+2ab+b²

Where , a = √3 and b = √2

=> y = 3+2√6+2

=> y = 5+2√6 --------------------(3)

On squaring both sides then

=> y² = (5+2√6)²

=> y² = 5²+2(5)(2√6)+(2√6)²

Since ,(a+b)² = a²+2ab+b²

Where , a = 5 and b = 2√6

=> y² = 25+20√6+24

=> y² = 49+20√6 ---------------(4)

On multiplying (1) and (3) then

xy = (5-2√6)(5+2√6)

=> xy = (5)²-(2√6)²

Since (a+b)(a-b) = a²-b²

Where , a = 5 and b = 2√6

=> xy = 25-24

=> xy = 1 -------------------------(5)

Now,

On adding (2),(4) and (5)

=> x²+y²+xy

=> 49-20√6+49+20√6+1

=> (49+49+1)+(20√6-20√6)

=> 99+0

=> 99

Therefore, x²+y²+xy = 99

Answer:-

The value of x²+y²+xy for the given problem is 99

Used formulae:-

→ (a+b)² = a²+2ab+b²

→ (a-b)² = a²-2ab+b²

→ (a+b)(a-b) = a²-b²

→ The Rationalising factor of √a+√b is √a-√b

→ The Rationalising factor of √a-√b is √a+√b

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