Question

if x = root 2 + 1/root2-1 and y = root2-1/root2+1 find the value of x + y+ xy

Answer 1

Answer:35

Step-by-step explanation:

Answer:for x, rationalise the denominator then we get 3+2root2

And for y rationalise the denominator then we get 3-2root2

Now, put the values of x and y in the polynomial, after the calculating we get 35. The answer is 35.

Answer 2

Step-by-step explanation:

Given :-

x = (√2 + 1)/(√2-1)

y = (√2-1)/(√2+1)

To find :-

Find the value of x +y+xy ?

Solution:-

Given that :

x = (√2 + 1)/(√2-1)

The denominator = √2-1

Rationalising factor of√2-1 = √2+1

On Rationalising the denominator then

=> x = [(√2 + 1)/(√2-1)]×[(√2+1)/(√2+1)]

=> x = [(√2+1)(√2+1)]/[(√2-1)(√2+1)]

=> x = (√2+1)²/[(√2-1)(√2+1)]

=> x = (√2+1)²/[(√2)²-1²)]

Since (a+b)(a-b) = a²-b²

Where a = √2 and b = 1

=> x = (√2+1)²/(2-1)

=> x =(√2+1)²/1

=> x = (√2+1)²

=> x = (√2)²+2(√2)(1)+(1)²

Since (a+b)² = a²+2ab+b²

Where a = √2 and b = 1

=> x = 2+2√2+1

=> x = 3+2√2----------------(1)

and

y = (√2-1)/(√2+1)

The denominator = √2+1

Rationalising factor of√2+1 = √2-1

On Rationalising the denominator then

=> y = [(√2 - 1)/(√2+1)]×[(√2-1)/(√2-1)]

=> y = [(√2-1)(√2-1)]/[(√2+1)(√2-1)]

=> y = (√2-1)²/[(√2+1)(√2-1)]

=> y = (√2-1)²/[(√2)²-1²)]

Since (a+b)(a-b) = a²-b²

Where a = √2 and b = 1

=> y = (√2-1)²/(2-1)

=> y =(√2-1)²/1

=> y = (√2-1)²

=> y = (√2)²-2(√2)(1)+(1)²

Since (a-b)² = a²-2ab+b²

Where a = √2 and b = 1

=> y = 2-2√2+1

=> y = 3-2√2----------------(2)

Now

On adding (1)&(2)

x+y = 3+2√2+3-2√2

=>x+y = 3+3

x+y = 6 ------------------(3)

On multiplying (1)&(2)

=> xy = (3+2√2)(3-2√2)

=> xy = (3)²-(2√2)²

Since (a+b)(a-b)=a²-b²

Where a = 3 and b = 2√2

=> xy = 9-8

=> xy = 1-----------------(4)

On adding (3)&(4)

x+y+xy = 6+1

x+y+xy = 7

Alternative Method :-

Given that

x = (√2 + 1)/(√2-1)

y = (√2-1)/(√2+1)

x+y+xy

=> (√2 + 1)/(√2-1)+(√2-1)/(√2+1)+(√2 + 1)/(√2-1)(√2-1)/(√2+1)

=> (√2 + 1)/(√2-1)+(√2-1)/(√2+1)+1

=>[[ (√2+1)²+(√2-1)²]/(√2+1)(√2-1)] +1

=> [(2+2√2+1+2-2√2+1)/(2-1)] +1

=> (2+1+2+1)+1

=> 6+1

=> 7

Answer:-

The value of x+y+xy for the given problem is 7

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