Solving Quadratic Equations

A **quadratic equation** is an equation of the form,

ax^{2} + bx + c = 0 ,

where a, b, and c are real numbers, with a ≠ 0 . The condition, a ≠ 0 , ensures
that the

equation actually does have a x^{2}-term. When solving quadratic equations, we
consider

two cases: b = 0 and b ≠ 0 .

When b = 0, a quadratic equation is the form of ax^{2} − c = 0 so we use the square

root property to quadratic equations when b = 0.

**Square Root Property
**

For any real number k, the equation x

If k > 0, then x

k < 0, then x

k = 0, then 0 is the only solution to x

Examples: Solving ax

Solutions:

(a) Use the square root property to solve x^{2} − 9 = 0 .

Thus, the solution set for the equation x^{2} − 9 = 0 is {-3, 3}.

(b) Using the square root property to solve, we get

Thus, the solution set for the equation
is
.

(c) Use the square root property to solve .

Because the square of any real number is nonnegative, the equation

has no real solution.

For the case when b ≠ 0 , we can solve quadratic equations by factoring,

completing the square, and using the quadratic formula.

**Solving Quadratic Equations by Factoring**

**Zero Factor Property**

If A and B are algebraic expressions, then the equation AB=0 is equivalent to
the

compound statement A = 0 or B = 0.

Examples:

Solve the following quadratic equations by factoring.

(a) x^{2} − x −12 = 0

(b) (x + 3)(x − 4) = 8

Solutions:

Using the zero factor property, we get

Thus, the solution set is {-3, 4} .

(b) (x + 3)(x − 4) = 8

First, multiply the left side using the FOIL method and subtract 8 from both sides.

Using the zero factor property, we get

Thus, the solution set is {-4, 5} .

**Solving Quadratic Equations by Completing the Square
**To complete the square of x

square of half the coefficient of x to both sides.

Examples:

Solve the following quadratic equations using completing the square

(a) x^{2} + 6x + 7 = 0

(b) 2x^{2} −3x − 4 = 0

Solutions:

(a) First, subtract 7 from both sides and then add
to both sides:

The solution set is .

(b) First, divide both sides by the leading coefficient, which is 2.

Now, add 2 to both sides and to both sides.

Thus, the solution set is .

**Solving Quadratic Equations using the Quadratic Formula**

**Quadratic Formula**

The solution to ax^{2} + bx + c = 0 , with a ≠ 0 , is given by the formula,

provided b^{2} − 4ac ≥ 0 . If b^{2} − 4ac < 0 , there are NO
REAL SOLUTIONS.

Example: Solve the quadratic equation, x^{2} + 8x + 6 = 0 , using the quadratic

formula.

Solution: Since a = 1, b = 8, and c = 6 ,

Thus, the solution set is .