Question

If the harmonic mean between roots of (5 + √2)x² - bx + 8 + 2√5 = 0 is 4 , then find the value of b

Answer 1

★ QUADRATIC RESOLUTION ★

Let α, β and be the roots of the equation , whose H.M. is 4

Then accordingly ,

4 = 2 αβ / α + β

4 = 2 × (8 + 2√5 / 5 + √2 )/ b / 5 + √2

2 = 8 + 2√5 / b

Hence , b = 4 + √5

★✩★✩★✩★✩★✩★✩★✩★✩★✩★✩★

Answer 2

Heya user,

Considering α, β as roots..

H.M. of roots = 4
=> 2 / [1/α + 1/β ] = 4
=> 1 / [ α + β ] / [ αβ ] = 2
=> [ αβ ] / [ α + β ] = 2
=> 2 [ α + β ] = [ αβ ] ---> ( i )

Now, by Roots Co-efficient relation,
α + β = b / [ 5 + √2 ] 

Putting this in (i) -->
--->  [ αβ ] = 2 b / [ 5 + √2 ]
==>  [ 5 + √2 ] [ αβ ] = 2 b; ---> ( ii )

Again, αβ = ( 8 + 2√5 ) / [ 5 + √2 ] ,  Putting this in (ii),
------->  [ 5 + √2 ] * ( 8 + 2√5 ) / [ 5 + √2 ] = 2 b
===> b = ( 8 + 2√5 ) / 2 = 4 + √5 <-- Desired result