4.8 Use the Quadratic Formula and the Discriminant

Quadratic formula

The formula that gives the solutions to any quadratic equation

Discriminant

The expression b² - 4ac under the radical sign of the quadratic formula

THE QUADRATIC FORMULA

Let a, b, and c be real numbers such that a ¹ 0. The solutions of the quadratic equation ax² + bx + c are:

-

±

-

2

4

2

b

b

ac

a

x =

Example 1

Solve an equation with two real solutions

Solve x² + 7x = 6.

x² + 7x = 6

Original equation

-

±

-

2

4

b

b

ac

x² + 7x _- 6_ = 0

Standard form

2

a

x =

Quadratic formula

x =

a = _1_, b = _7_, c = _-6_

x =

Simplify.

The solutions are x = » _0.77_ and x = » _-7.77_.

Example 2

Solve an equation with one real solution

Solve 2x² - 8x + 8 = 0.

Solution

2x^2 - 8x + 8 = 0	Original equation
x =	a = _2_, b = _-8_, c = _8_
x =	Simplify.
x = _2_	Simplify.
The solution is _2_.

Example 3

Solve an equation with imaginary solutions

Solve -x² + 2x = 5.

Solution

-x^2 + 2x = 5	Original equation
-x^2 + 2x _-5_ = 0	Standard form
x =	a = __-1_, b = _2_, c = _-5_
x =	Simplify.
x =	Rewrite using the imaginary unit i.
x = _1 ± 2i_	Simplify.

The solutions are _1 + 2i_ and _1 - 2i_.

USING THE DISCRIMINANT OF ax² + bx + c = 0

When b² - 4ac > 0, the equation has _two real solutions_. The graph has _two_

x-intercepts.

When b² - 4ac = 0, the equation has _one real solution_. The graph has _one_

x-intercept.

When b² - 4ac < 0, the equation has _two imaginary solutions_. The graph has _no_

x-intercepts.

Example 4

Use the discriminant

Find the discriminant of the quadratic equation and give the number and type of solutions of the equation.

a. x² + 6x + 5 = 0

b. x² + 6x + 9 = 0

c. x² + 6x + 13 = 0

Discriminant	Solution(s)
b^2 - 4ac	x =
a. _(6)^2 - 4(1)(5) =16____	_Two real: -5, -1______
b. _(6)^2 - 4(1)(9) = 0____	_One real: -3__________
c. _(6)^2 - 4(1)(13) = -16_	_Two imaginary: -3 ± 2i_