Quadratic equations – derivation of quadratic equation formula using completing-the-square method

We discussed earlier that the general form of a quadratic equation is given by the relation

ax² + bx + c = 0

where a ≠ 0.

The reason why a must not be zero is that if ‘a’ = 0, the equation will no longer be quadratic. Click here to revise the lesson.

This general form is solved to derive the quadratic formula shown below 

The method of doing this is called completing-the-square method.

Now, let us solve and show how this is done.

Given that ax² + bx + c = 0

Step1

Subtract c from both sides of the equation as follows

ax² + bx + c - c = 0 – c

ax² + bx = - c

Note: You can watch the video that explains this derivation by clicking here

Step2

Make the coefficient of x² to be 1. To do this, divide through by a (the current coefficient of x²) 

Note: You can watch the video that explains this derivation by clicking here

Note: You can watch the video that explains this derivation by clicking here