Question

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

x² – 2ax – (4b² – a²) = 0

Answer 1

The given equation is x² – 2ax – (4b² – a²) = 0 

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

A = 1, B = –2a and C = –(4b² – a²) 

∴ Discriminant,

B² – 4AC = (–2a)² – 4 × 1 × [–(4b² – a²)]

= 4a² + 16b² – 4a²

= 16b² > 0 

So, the given equation has real roots.

Now, `sqrt(D) = sqrt(16)b^2 = 4b`  

∴ `α = (-B + sqrt(D))/(2A)`

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

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

= a + 2b 

`β = (-B - sqrt(D))/(2A)`

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

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

= a – 2b 

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