if x = √5-√3/√5+√3 and y =√5+√3/√5-√3 find the value of x2 +y2
Answers
Step-by-step explanation:
Given :-
x = √5-√3/√5+√3 and
y =√5+√3/√5-√3
To Find:-
Find the value of x²+y²?
Solution:-
Given that :
x = (√5-√3)/(√5+√3)
The denominator = √5+√3
The Rationalising factor of√5+√3 is √5-√3
On Rationalising the denominator then
=> x=[(√5-√3)/(√5+√3)]×[(√5-√3)/(√5-√3)]
=> x = [(√5-√3)(√5-√3)]/[(√5+√3)(√5-√3)]
=> x = (√5-√3)²/[(√5+√3)(√5-√3)]
=> x = (√5-√3)²/[(√5)²-(√3)²]
Since (a+b)(a-b) = a² - b²
Where , a = √5 and b =√3
=> x = (√5-√3)²/(5-3)
=> x = (√5-√3)²/2
=> x = [(√5)²-2(√5)(√3)+(√3)²]/2
Since (a-b)² = a²-2ab+b²
=> x = (5-2√15+3)/2
=> x = (8-2√15)/2
=> x = 2(4-√15)/2
=> x = (4-√15)/1
=> x = 4-√15
On squaring both sides then
=> x² = (4-√15)²
=> x² = 4²-2(4)(√15)+(√15)²
Since (a-b)² = a²-2ab+b²
=> x² = 16-8√15+15
=> x² = 31-8√15 -----------------(1)
and
given that
y = (√5+√3)/(√5-√3)
The denominator = √5-√3
The Rationalising factor of√5-√3 is √5+√3
On Rationalising the denominator then
=>y=[(√5+√3)/(√5-√3)]×[(√5+√3)/(√5+√3)]
=> y = [(√5+√3)(√5+√3)]/[(√5-√3)(√5+√3)]
=> y = (√5+√3)²/[(√5+√3)(√5-√3)]
=> y = (√5+√3)²/[(√5)²-(√3)²]
Since (a+b)(a-b) = a² - b²
Where , a = √5 and b =√3
=> y = (√5+√3)²/(5-3)
=> y = (√5+√3)²/2
=> y = [(√5)²+2(√5)(√3)+(√3)²]/2
Since (a+b)² = a²+2ab+b²
=> y = (5+2√15+3)/2
=> y = (8+2√15)/2
=> y = 2(4+√15)/2
=> y = (4+√15)/1
=> y = 4+√15
On squaring both sides then
=> y² = (4+√15)²
=> y² = 4²+2(4)(√15)+(√15)²
Since (a+b)² = a²+2ab+b²
=> y² = 16+8√15+15
=> y² = 31+8√15 -----------------(2)
On adding (1) & (2) then
x²+y² = (31-8√15)+(31+8√15)
=> x²+y² = (31+31)+(8√15-8√15)
=> x²+y² = 62+0
=> x²+y² = 62
Answer:-
The value of x²+y² for the given problem is 62
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