Using the different methods if (applicable) solve the following quadratic equations:?

Question

3x^2 + x - 1 = 0

Explain which method (or methods) may not be applicable at all times and why. Is the solution a set of real numbers, imaginary numbers, or complex numbers?

Answer 1

Using quadratic formula , which always works:-
x = [ - 1 ± √(1 + 12) ] / 6
x = [ - 1 ± √(13) ] / 6
x = 0.434 , x = - 0.768

Factorising does not work in this case.

Completing the square leads to the quadratic formula:-
3x² + x - 1 = 0
3 [x² + 1 / 3 x - 1 / 3] = 0
3 [ (x² + 1 / 3 x + 1 / 36) - 1 / 36 - 1 / 3 ] = 0
3 [(x + 1 / 6)² - 13 / 36 ] = 0
(x + 1 / 6)² = 13 / 36
x + 1 / 6 = ± √(13) / 6
x = [ -1 ± √(1 + 12)] / 6
x = [- 1 ± √(13)] / 6
Which leads back to answer as obtained by quadratic formula.

Answer 2

The Quadratic Formula will work for any quadratic equation.
The Box Method is also applicable here.
Graphically solving it is also a possibility, but this will be the least accurate answer.