Question

Find the roots of the following equation, if they exist, by applying the quadratic formula:

36x² – 12ax + (a² – b²) = 0 

Answer 1

The given equation is 36x² – 12ax + (a² – b²) = 0  

Comparing it with Ax² + Bx + C = 0 

A = 36, B = –12a and C = a² – b² 

∴ Discriminant, D = B² – 4AC

= (–12a)² – 4 × 36 × (a² – b²)

= 144a² – 144a² + 144b²

= 144b² > 0 

So, the given equation has real roots.

Now, `sqrt(D) = sqrt(144b^2) = 12b` 

∴ `α = (-B + sqrt(D))/(2A)`

= `(-(-12a) + 12b)/(2 xx 36)`

= `(12(a + b))/72`

= `(a + b)/0`

`β = (-B - sqrt(D))/(2A)`

= `(-(-12a) - 12b)/(2 xx 36)`

= `(12(a - b))/72`

= `(a - b)/6` 

Hence, `(a + b)/6` and `(a - b)/6` are the roots of the given equation.