Find the solutions to this quadratic equation: 0 = 2x^2 - 5x - 3

First we need to factorise this quadratic. This will be in the form:

0 = (Ax + B)(Cx + D)
You can work this out the values of A, B, C and D by looking at the coefficients in the unfactorised form. For example, you know that A x C = 2 as this is the coefficient of x^2, therefore the only two options for A and C are 1 and 2. You also know that B x D = -3, so you know that B and D must be +/-3 and +/-1. From here you can use trial and error to see with value is which.
0 = (2x + 1)(x - 3)
We know that for this to equal zero, either (2x + 1)=0, or (x-3)=0, so x=-0.5 and x=3