Question

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

`2x^2 + 6sqrt(3)x - 60 = 0`

Answer 1

The given equation is `2x^2 + 6sqrt(3)x - 60 = 0`

Comparing it with ax² + bx + c = 0 

a = 2, b = `6sqrt(3)` and c = –60 

∴ Discriminant, D = b² – 4ac 

= `(6sqrt(3))^2 - 4 xx 2 xx (-60)`

= 180 + 480

= 588 > 0  

So, the given equation has real roots.

Now, `sqrt(D) = sqrt(588) = 14sqrt(3)` 

∴ `α = (-b + sqrt(D))/(2a)`

= `(-6sqrt(3) + 14sqrt(3))/(2 xx 2)`

= `(8sqrt(3))/4`

= `2sqrt(3)` 

`β = (-b + sqrt(D))/(2a)`

= `(-6sqrt(3) + 14sqrt(3))/(2 xx 2)`

= `(-20sqrt(3))/4`

= `-5sqrt(3)`

Hence, `2sqrt(3)` and `-5sqrt(3)` are the roots of the given equation.