Exer. Solve by completing the square. (Note: See the discussion after Example for help in...

Exer. Solve by completing the square. (Note: See the discussion after Example for help in solving Exercises)

EXAMPLE Solving a quadratic equation by completing the square

Solve the equation x2 – 5x + 3 = 0.

SOLUTION It is convenient to first rewrite the equation so that only terms involving x are on the left-hand side, as follows:

Thus, the solutions of the equation are  and

In Example, we solved a quadratic equation of the form ax2 + bx + c = 0 with a = 1. If a ≠ 1, we can solve the quadratic equation by adding a step to the procedure used in the preceding example. After rewriting the equation so that only terms involving x are on the left-hand side,

we divide both sides by a, obtaining

We then complete the square by adding  to both sides. This technique is used in the proof of the following important formula.

4x2 − 12x − 11 = 0