Question

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

`2x^2 + 5sqrt(3)x + 6 = 0`

Answer 1

Given: `2x^2 + 5sqrt(3)x + 6 = 0`

Step-wise calculation:

1. Compare with ax² + bx + c = 0:

a = 2, b = `5sqrt(3)`, c = 6

2. Quadratic formula:

`x = (-b ± sqrt(b^2 - 4ac))/(2a)`

3. Discriminant:

D = b² – 4ac 

= `(5sqrt(3))^2 - 4 xx 2 xx 6` 

= 75 – 48

= 27

4. `sqrt(D) = sqrt(27)`

= `3sqrt(3)`

5. `x = (-5sqrt(3) ± 3sqrt(3))/(2 xx 2)` 

= `(-5sqrt(3) ± 3sqrt(3))/4`

`x_1 = (-5sqrt(3) + 3sqrt(3))/4`

= `(-2sqrt(3))/4` 

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

`x_2 = (-5sqrt(3) - 3sqrt(3))/4` 

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

= `-2sqrt(3)`

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