Question

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

x² + 5x – (a² + a – 6) = 0

Answer 1

The given equation is x² + 5x – (a² + a – 6) = 0

Comparing it with Ax² + Bx + C = 0 

A = 1, B = 5 and C = –(a² + a – 6) 

∴ Discriminant, D = B² – 4AC

= 5² – 4 × 1 × [–(a² + a – 6)]

= 25 + 4a² + 4a – 24

= 4a² + 4a² + 4a + 1 

= (2a + 1)² > 0  

So, the given equation has real roots.

Now, `sqrt(D) = sqrt((2a + 1)^2) = 2a + 1` 

∴ `α = (-B + sqrt(D))/(2A)`

= `(-5 + 2a + 1)/(2 xx 1)`

= `(2a - 4)/2`

= a – 2

`β = (-B - sqrt(D))/(2A)`

= `(-5 - (2a + 1))/(2 xx 1)`

= `(-2a - 6)/2`

= –a – 3

= –(a + 3) 

Hence, (a – 2) and –(a + 3) are the roots of the given equation.