Question

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

x² – (2b – 1)x + (b² – b – 20) = 0

Answer 1

The given equation is x² – (2b – 1)x + (b² – b – 20) = 0

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

A = 1, B = –(2b – 1) and C = b² – b – 20 

∴ Discriminant, D = B² – 4AC

= [–(2b – 1)]² – 4 × 1 × (b² – b – 20)

= 4b² – 4b + 1 – 4b² + 4b + 80

= 81 > 0 

So, the given equation has real roots.

Now, `sqrt(D) = sqrt(18) = 9` 

∴ `α = (-B + sqrt(D))/(2A)`

= `(-[-(2b-1)] + 9)/(2 xx 1)`

= `(2b + 8)/2`

= b + 4  

`β = (-B - sqrt(D))/(2A)`

= `(-[-(2b - 1)] - 9)/(2 xx 1)`

= `(2b - 10)/2`

= b – 5  

Hence, (b + 4) and (b – 5) are the roots of the given equation.