Solve the equation 4x^2-12x=7 algebraically for x.

                            4x² - 12x = 7
                                      - 7  - 7
                       4x² - 12x - 7 = 0
                4x² + 2x - 14x - 7 = 0
2x(2x) + 2x(1) - 7(2x) - 7(1) = 0
          2x(2x + 1) - 7(2x + 1) = 0
                    (2x - 7)(2x + 1) = 0
       2x - 7 = 0    or    2x + 1 = 0
           + 7 + 7                 - 1  - 1
             2x = 7                 2x = -1
              2     2                  2      2
               x = 3¹/₂      or       x = ⁻¹/₂

First subtract 7 from each side of the equation.

4x^2 - 12x - 7 = 0.

Now we can substitute terms into our friend the quadratic equation, which you should memorize.  Or we can try a few simple factors.  7 is prime, so to make -7 one number must be positive and one must be negative.

The first pair I tried, (2x + 1)(2x - 7) yield a middle term of -12x.  Therefore, setting each to zero yields:  x = -1/2, or x = 7/2.