# Quadratic Equations

**Solving by Factoring**

A quadratic equation is an equation that can be written in the form

ax^{2} + bx + c = 0

where a, b, and c are real numbers with a ≠ 0.

To solve a quadratic equation by factoring, rewrite the
equation, if necessary, so that one

side is equal to 0 and use the Zero-Product Property:

ab = 0 if and only if a = 0 or b = 0.

**Example 1:** Solve the following equations by factoring.

**Example 2:** Solve 9x^{2} −16 = 0 using the square root
method.

**Solving by Completing the Square**

Given x^{2} + bx + c = 0

1. Rewrite the equation as x^{2} + bx = −c (Notice that the
leading coefficient is positive 1,

if it’s not then you will have to divide both sides of the equation by the
leading

coefficient.) and make the left –hand side a perfect square.

2. Make the left-hand side a perfect square by adding
to
both sides (to balance the

equation)

3. Factor the left-hand side.

4. Use the square root property to solve.

**Example 3:** Find all real solutions of the following
equations by completing the square.

**Solving by the Quadratic Formula**

The solutions of the equation ax^{2} + bx + c = 0 , where a
≠ 0 , can be found by using the

quadratic formula:

**Example 4: **Find all real solutions of 3x^{2} + 2x + 2 = 0 by
using the quadratic formula.

Note: The discriminant of the equation ax^{2} + bx + c = 0 ( a ≠ 0) is given by

D = b^{2} − 4ac .

If D > 0, then the equation ax^{2} + bx + c = 0 has two
distinct real solutions.

If D = 0, then the equation ax^{2} + bx + c = 0 has exactly one real solution.

If D < 0, then the equation ax^{2} + bx + c = 0 has no real solution (The roots of
the

equation are complex numbers and appear as complex conjugate pairs.)