Question

Find the product of (3+√5) and (3-√5)

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Answer 1

Answer:

4

Explanation:

It is in the form of (a+b) (a-b) =a2-b2

=9-5

=4

Answer 2

Answer:

3+5–√3−5–√=a+b5–√3+53−5=a+b5

Take the conjugate of the denominator term and multiply both numerator and denominator by that conjugate.

Here, conjugate of 3−5–√3−5 is 3+5–√3+5 .

Now,

(3+5–√3−5–√)×(3+5–√3+5–√)=a+b5–√(3+53−5)×(3+53+5)=a+b5

⟹(3+5–√)2(3−5–√)(3+5–√)=a+b5–√⟹(3+5)2(3−5)(3+5)=a+b5

We know

(a+b)2=a2+b2+2ab⋯(1)(a+b)2=a2+b2+2ab⋯(1)

(a+b)(a−b)=a2−b2⋯(2)(a+b)(a−b)=a2−b2⋯(2)

Apply formula (1) in the numerator and formula (2) in the denominator.

(3)2+(5–√)2+2×3×5–√32−(5–√)2=a+b5–√(3)2+(5)2+2×3×532−(5)2=a+b5

9+5+65–√9−5=a+b5–√9+5+659−5=a+b5

⟹14+65–√4=a+b5–√⟹14+654=a+b5

Take 2 as common from the numerator.

2(7+35–√)4=a+b5–√2(7+35)4=a+b5

⟹7+35–√2=a+b5–√⟹7+352=a+b5

⟹72+325–√=a+b5–√⟹72+325=a+b5

Comparing the corresponding rational and irrational parts on both sides, we get:

a=72,b=32a=72,b=32

or

a=3.5,b=1.5a=3.5,b=1.5