Question

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

3x² – 2x + 2 = 0

Answer 1

Given: 3x² – 2x + 2 = 0

Step-wise calculation:

1. Compare with ax² + bx + c = 0:

a = 3, b = –2, c = 2

2. Quadratic formula:

`x = (-b ± sqrt(b^2 - 4ac))/(2a)`

3. Discriminant:

D = b² – 4ac 

= (–2)² – 4(3)(2) 

= 4 – 24

= –20

4. Since D < 0 there are no real roots; the roots are complex.

5. Compute complex roots:

`x = (-(-2) ± sqrt(-20))/(2 xx 3)` 

= `(2 ± isqrt(20))/6` 

`sqrt(20) = 2sqrt(5)` 

So, `x = (2 ± 2isqrt(5))/6`

= `(1 ± isqrt(5))/3`

The equation has no real roots. Its two complex roots are `x = (1 + isqrt(5))/3` and `x = (1 - isqrt(5))/3`.