Question

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

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

Answer 1

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

Step-wise calculation:

1. Compare with ax² + bx + c = 0:

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

2. Discriminant:

D = b² – 4ac 

= `(-2sqrt(6))^2 - 4(3)(2)`

= 4 × 6 – 24 

= 24 – 24

= 0

3. Quadratic formula:

`x = (-b ± sqrt(D))/(2a)`

Because D = 0, both roots are equal: `x = (-b)/(2a)`.

4. Compute the root:

`x = -(-2sqrt(6))/(2 xx 3)` 

= `(2sqrt(6))/6` 

= `sqrt(6)/3`

The equation has one repeated real root (multiplicity 2): `x = sqrt(6)/3`.