Question

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

`sqrt(3)x^2 - 2sqrt(2)x - 2sqrt(3) = 0`

Answer 1

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

Step-wise calculation:

1. Identify coefficients:

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

2. Discriminant:

D = b² – 4ac

`b^2 = (-2sqrt(2))^2`

= 8

`-4ac = -4(sqrt(3))(-2sqrt(3))` 

= +24

So, D = 8 + 24 = 32.

3. `sqrt(D) = sqrt(32)`

= `4 sqrt(2)`

4. Quadratic formula:

`x = (-b ± sqrt(D))/(2a)` 

`-b = 2sqrt(2)` 

`2a = 2 sqrt(3)` 

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

= `(2sqrt(2)(1 ± 2))/(2sqrt(3))`

= `(sqrt(2)(1 ± 2))/sqrt(3)`

5. Compute each root:

For +: `x_1 = (sqrt(2) xx 3)/sqrt(3)` 

= `(3sqrt(2))/sqrt(3)` 

= `sqrt(6)` 

For -: `x_2 = (sqrt(2) xx (-1))/sqrt(3)` 

= `-sqrt(2)/sqrt(3)` 

= `-sqrt(6)/3`

The roots are `x = sqrt(6)` and `x = -(1/3)sqrt(6)`.