Question

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

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

Answer 1

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

Step-wise calculation:

1. Identify coefficients:

a = `2sqrt(3)`, b = –5, c = `sqrt(3)`

Use the quadratic formula `x = (-b ± sqrt(b^2 - 4ac))/(2a)`.

2. Compute the discriminant:

Δ = b² – 4ac 

= `(-5)^2 - 4(2sqrt(3))(sqrt(3))` 

= 25 – 4(2 × 3) 

= 25 – 24

= 1

3. Take square root:

`sqrt(Δ) = 1`

4. Apply the formula:

`x = (-(-5) ± 1)/(2(2sqrt(3)))` 

= `(5 ± 1)/(4sqrt(3))`

5. Compute each root and simplify:

`x_1 = (5 + 1)/(4sqrt(3))` 

= `6/(4sqrt(3))` 

= `3/(2sqrt(3))` 

= `sqrt(3)/2`   ...(Rationalized)

`x_2 = (5 - 1)/(4sqrt(3))` 

= `4/(4sqrt(3))` 

= `1/sqrt(3)`

= `sqrt(3)/3`   ...(Rationalized)

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