Question

Determine the nature of the roots of the following quadratic equation:

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

Answer 1

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

Step-wise calculation:

1. Identify coefficients:

`a = 4, b = 4sqrt(3), c = 3`

2. Discriminant: D = b² – 4ac 

= `(4sqrt(3))^2 - 4 xx 4 xx 3` 

= 16 × 3 – 48 

= 48 – 48

= 0

3. Since D = 0, the quadratic has two real and equal (repeated) roots (discriminant rule).

4. Root value: `x = (-b)/(2a)` 

= `(-(4sqrt(3)))/(2 xx 4)` 

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

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

The equation has two equal real roots; both roots are `x = -sqrt(3)/2`.