Question

Solve for x, if `5/x + 4sqrt(3) = (2sqrt(3))/x^2, x = 0`

Answer 1

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

Step-wise calculation:

1. Multiply both sides by x² valid since x ≠ 0:

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

2. Rearrange to standard quadratic form:

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

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

Compute the discriminant:

Δ = b² – 4ac 

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

= 25 + 96

= 121

4. `sqrt(Δ) = 11`.

Apply the quadratic formula:

`x = (-b ± sqrt(Δ))/(2a)`

= `(-5 ± 11)/(8sqrt(3))`

5. Compute the two roots:

`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 solutions are `x = sqrt(3)/4` and `x = (-2sqrt(3))/3` both nonzero, so both valid.