Question

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

x² + 6x – (a² + 2a – 8) = 0

Answer 1

The given equation is x² + 6x – (a² + 2a – 8) = 0

Comparing it with Ax² + Bx + C = 0 

A = 1, B = 6 and C = –(a² + 2a – 8) 

∴ Discriminant, D = B² – 4AC

= 6² – 4 × 1 × [–(a² + 2a – 8)]

= 36 + 4a² + 8a – 32

= 4a² + 8a – 32

= 4a² + 8a + 4 

= 4(a² + 2a + 1)

= 4(a + 1)² > 0 

So, the given equation has real roots.

Now, `sqrt(D) = sqrt(4(a + 1)^2) = 2(a + 1)`  

∴ `α = (-B + sqrt(D))/(2A)`

= `(-6 + 2(a + 1))/(2 xx 1)`

= `(2a - 4)/2`

= a – 2 

`β = (-B - sqrt(D))/(2A)`

= `(-6 - 2(a + 1))/(2 xx 1)`

= `(2a - 8)/2`

= a – 4

= –(a + 4) 

Hence, (a – 2) and –(a + 4) are the roots of the given equation.