Question

if x = √3+√2, then x²+1/x² = ??​

Answer 1

The value of x²+(1/x²) is 10

Given :-

x = √3+√2

To find :-

The value of x²+(1/x²)

Solution :-

Given that

x = √3+√2 ------------(1)

=> 1/x = 1/(√3+√2)

The denominator = √3+√2

We know that

The Rationalising factor of √a+√b = √a-√b

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

On Rationalising the denominator then

1/x = [1/+(√3+√2)]×[(√3-√2)/(√3-√2)]

=> 1/x = [1(√3-√2)]/[(√3+√2)(√3-√2)]

=> 1/x = (√3-√2)/([(√3)²-(√2)²]

Since, (a+b)(a-b) = a²-b²

Where, a = √3 and b = √2

=> 1/x = (√3-√2)/(3-2)

=> 1/x = (√3-√2)/1

=> 1/x = √3-√2 -------(2)

We know that

(a+b)² = a²+2ab+b²

Now,

[x+(1/x)]² = x²+2(x)(1/x)+(1/x)²

=> [x+(1/x)]² = x²+2(x/x)+(1/x²)

=> [x+(1/x)]² = x²+2(1)+(1/x²)

=> [x+(1/x)]² = x²+(1/x²)+2

=> x²+(1/x²) = [x+(1/x)]²-2

=> x²+(1/x²) = (√3+√2+√3-√2)²-2

=> x²+(1/x²) = (√3+√3)²-2

=> x²+(1/x²) = (2√3)²-2

=> x²+(1/x²) = 12-2

Therefore, x²+(1/x²) = 10

Answer :-

The value of x²+(1/x²) is 10

Used formulae:-

• (a+b)² = a²+2ab+b²

• (a+b)(a-b) = a²-b²

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