Question

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

x² – 4ax – b² + 4a² = 0

Answer 1

The given equation is x² – 4ax – b² + 4a² = 0

Comparing it with Ax² + Bx + C = 0, we get

A = 1, B = –4a and C = –b² + 4a²

∴ Discriminant, D = B² – 4AC

= (–4a)² – 4 × 1 × (–b² + 4a²)

= 16a² + 4b² – 16a²

= 4b² > 0 

So, the given equation has real roots.

Now, `sqrt(D) = sqrt(4b^2) = 2b` 

∴ `α = (-B + sqrt(D))/(2A)`

= `(-(-4a) + 2b)/(2 xx 1)`

= `(4a + 2b)/2`

= 2a + b 

`β = -(-B - sqrt(D))/(2A)`

= `(-(-4a) - 2b)/(2 xx 1)`

= `(4a - 2b)/2`

= 2a – b 

Hence, (2a + b) and (2a – b) are the roots of the given equation.