Question

Solve the following quadratic equation:

`sqrt(3)x^2 + 10x - 8sqrt(3) = 0`

Answer 1

Given: `sqrt(3)x^2 + 10x - 8sqrt(3) = 0`.

Identify coefficients: `a = sqrt(3), b = 10, c = -8sqrt(3)`.

Discriminant: Δ = b² – 4ac 

= `10^2 - 4(sqrt(3))(-8sqrt(3))` 

= 100 – 4(–24)

= 100 + 96

= 196

`sqrt(Δ) = 14`

Quadratic formula: `x = (-b ± sqrt(Δ))/(2a)`

= `(-10 ± 14)/(2sqrt(3))`

`x_1 = (-10 + 14)/(2sqrt(3))`

= `4/(2sqrt(3))`

= `2/sqrt(3)`

= `(2sqrt(3))/3`   ...(Rationalized)

`x_2 = (-10 - 14)/(2sqrt(3))`

= `(-24)/(2sqrt(3))`

= `(-12)/sqrt(3)`

= `-4sqrt(3)`

The roots are `x = 2/sqrt(3) = (2sqrt(3))/3` and `x = −4sqrt(3)`.