Question

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

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

Answer 1

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

Step-wise calculation:

1. Compare with ax² + bx + c = 0: 

a = `sqrt(3)`, b = 10, c = `-8sqrt(3)`

2. Discriminant:

D = b² – 4ac 

= `100 - 4(sqrt(3))(-8sqrt(3))` 

= 100 + 96

= 196

3. `sqrt(D) = 14`

4. Quadratic formula:

`x = (-b ± sqrt(D))/(2a)` 

= `(-10 ± 14)/(2sqrt(3))` 

`x_1 = (-10 + 14)/(2sqrt(3))` 

= `4/(2sqrt(3))` 

= `2/sqrt(3)` 

= `(2sqrt(3))/3` 

`x_2 = (-10 - 14)/(2sqrt(3))` 

= `(-24)/(2sqrt(3))` 

= `(-12)/sqrt(3)` 

= `-4sqrt(3)`

The equation has two real roots: `x = (2sqrt(3))/3` and `x = -4sqrt(3)`.