# Quadratic Equations

**Definition. ** A** quadratic equation** in x is an
equation that can be written in the form

ax^{2} + bx + c = 0, where a ≠ 0.

Examples: x^{2} + x = 2, x^{2} + 6x + 5 = 0.

**Zero Factor Theorem.** If p and q are algebraic expressions, then

pq = 0 if and only if p = 0 or q = 0.

**Example 3.1.** Solving an equation by factoring.

Solve the equation x^{2} + x = 2.

Note: To use the method of factoring, it is essential that only the number 0
appear on one

side of the equation.

x^{2} + x = 2 subtract 2

x^{2} + x - 2 = 0 factor

(x - 1)(x + 2) = 0 zero factor theorem

x - 1 = 0 or x + 2 = 0

x = 1 or x = -2

So the solutions to the given equation are 1 and -2.

**Example 3.2**. Solving an equation by completing the square.

Solve the equation x^{2} + 6x + 5 = 0.

It is convenient to write the equation such that only terms involving x are on
the right.

Thus we have

x^{2} + 6x = -5.

We are trying to find a square of the type (x + a)^{2} such that when
expanded it contains

the terms x^{2}, 6x and a number. In this case, the square is (x + 3)^{2},
which expands as

x^{2} + 6x + 9. So to complete the square we add 9 both sides and we
obtain the following

equivalent equations:

**Quadratic Formula .** If a ≠ 0, then the roots of ax^{2}
+ bx + c = 0 are given by

Note: If the discriminant b^{2} - 4ac is negative, the equation does not
have real roots, but

we will learn in the next section that if r is a positive number then we can use
complex

numbers to define the principal square root of -r by
, where i is a special

complex number with the property that i^{2} = -1.

**Example 3.3.** Solving an equation using the quadratic formula.

Solve the equation 2x^{2} - 6x + 3 = 0.

We have a = 2, b = -6, c = 3. Thus