Question

find the harmonic mean of the roots of the equation

(7 + √2)x² - (6 + √7)x + 18 + 3√7 = 0   

1. 6
2. 4
3. 5
4. 8

Answer 1

Correct Answer - Option 1 : 6

Given∶

Quadratic equation, (7 + √2)x² - (6 + √7)x + 18 + 3√7 = 0  

Formula  Used∶

Harmonic mean = 2 α β /α + β 

Relationship between coefficient and zeros of the quadratic equation.

α + β  = - b/a, α β = c/a

Calculation∶

The given equation is 

(7 + √2)x2 - (6 + √7)x + 18 + 3√7 = 0       (1)

Let it roots p and q. So,

p + q = - b/a = -[-(6 + √7)]/(7 + √2) = (6 + √7)/(7 + √2) = (6 + √7)/(7 + √2)

and pq = c/a = (18 + 3√7)/(7 + √2)

We know that the harmonic mean of numbers p and q will be    

HM = 2pq/(p + q)      (2)

put the values of pq and p + q in eq.(2), we get

HM = 2 [(18 + 3√7)/(7 + √2)] / [(6 + √7)/(7 + √2)]

HM = 2 [(18 + 3√7) / (6 + √7)]

HM = 2 [3(6 + √7)/(6 + √7)]

HM = 2 × 3 = 6

∴ The required harmonic mean is 6.