Question

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

`4sqrt(3)x^2 + 5x - 2sqrt(3) = 0`

Answer 1

Given: `4sqrt(3)x^2 + 5x - 2sqrt(3) = 0`

Step-wise calculation:

1. Identify coefficients:

a = `4sqrt(3)`, b = 5, c = `-2sqrt(3)`

2. Discriminant:

D = b² – 4ac 

= `25 - 4(4sqrt(3))(-2sqrt(3))`

= 25 – (–96)

= 121

3. `sqrt(D) = 11`.

4. Quadratic formula:

`x = (-b ± sqrt(D))/(2a)` 

= `(-5 ± 11)/(8sqrt(3))`

5. Compute each root:

`x_1 = (-5 + 11)/(8sqrt(3))`

= `6/(8sqrt(3))`

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

= `sqrt(3)/4`

`x_2 = (-5 - 11)/(8sqrt(3))` 

= `(-16)/(8sqrt(3))` 

= `-2/sqrt(3)`

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

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