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