If a and b are rational numbers, find the values of a and b:
(√5- 2)/( √5+2) - (√5 + 2)/(√5 - 2) = a + b √5
PLS FIND THE ANSWER ASAP!!
Answers
Answer:
- 8√5
Step-by-step explanation:
Solution is given in figure.
Step-by-step explanation:
Given :-
a and b are rational numbers
(√5- 2)/( √5+2) - (√5 + 2)/(√5 - 2) = a + b √5
To find :-
Find the values of a and b?
Solution :-
Given that :
(√5- 2)/( √5+2) - (√5 + 2)/(√5 - 2)
= a + b √5
LHS = (√5- 2)/( √5+2) - (√5 + 2)/(√5 - 2)
RHS = a + b √5
On taking LHS
(√5- 2)/( √5+2) - (√5 + 2)/(√5 - 2)
Rationalising factor of √5+2 is √5-2
Rationalising factor of √5-2 is √5+2
On Rationalising the denominators then
=>[(√5-2)(√5-2)/(√5+2)(√5-2)] -
[(√5+2)(√5+2)/(√5-2)(√5+2)]
=> [(√5-2)²/(√5+2)(√5-2)]-[(√5+2)²/(√5-2)(√5+2)]
=> [(√5-2)²/[(√5)²-(2)²]]-[(√5+2)²/[(√5)²-2²]]
Since (x+y)(x-y) = x²-y²
Where , x =√5 and y = 2
=>[(√5-2)²/(5-4)]-[(√5+2)²/(5-4)]
=> [(√5-2)²/1]-[(√5+2)²/1]
=> (√5-2)²-(√5+2)²
=>[(√5)²-2(√5)(2)+2²]-[(√5)²+2(√5)(2)+2²]
Since (x+y)² = x²+2xy+y²
and (x-y)² = x²-2xy+y²
=> (5-4√5+4)-(5+4√5+4)
=> (9-4√5)-(9+4√5)
=> 9-4√5-9-4√5
=> (9-9)+(-4√5-4√5)
=> 0-8√5
=> -8√5
=> LHS = -8√5
Now LHS = RHS
-8√5 = a+b√5
It can be written as
0+(-8)√5 = a+b√5
On comparing both sides then
a = 0 and b = -8
Answer:-
The value of a = 0
The value of b = -8
Used formulae:-
- (x+y)² = x²+2xy+y²
- (x-y)² = x²-2xy+y²
- (x+y)(x-y) = x²-y²