MAT 108 Quadratic Equations

By

Dr. Marinas

Solving Equations

• In sections 1.1 and 1.2, we solved LINEAR equations. This means that the highest exponent is 1.

• In section 1.3, we will be solving equations with the highest exponent of 2. These are called Quadratic Equations.

What are Quadratic Equations?

• They are in the form: ax2 + bx + c = 0

• Methods of Solving Quadratic Equations (Put into above form first)?– By Factoring (sometimes works)– By Quadratic Formula (always works)

By Factoring

• Put in Standard Form

• Factor the polynomial• Set each factor to 0

• Solve for the variable• Check answer in

original equation

• x2 - 3x = 4 x2 - 3x - 4 = 0• (x - 4) (x + 1) = 0• x - 4 = 0 or x + 1 = 0• x = 4 or x = -1• 42 - 3(4) = 4 (-1)2 - 3(-1) = 4

By Quadratic Formula

• This method works all the time.

• It is usually used when the Factoring method fails.

• Put in standard form.• Determine the values of a,

b, and c.

The Quadratic Formula

_______ -b + b2 - 4ac• x = ------------------------ 2 a

Example:

• x2 + 9x = -8• Standard form: x2 + 9x + 8 = 0• We could do this by Factoring but let’s

use the Quadratic Formula.• a = 1, b = 9 and c = 8 -9 + 92 - 4(1)(8) -9 + 49• x = ------------------------ = ----------- 2(1) 2

Example Continues

-9 + 49 -9 + 7x = -------------- = ------- 2 2• x = (-9-7)/2 or x = (-9 + 7)/2• x = -8 or x = -1• x2 + 9x = -8• Check: (-8)2 + 9(-8) = -8 (-1)2 + 9(-1) = -8

Homework Pages 47-48

• Problems (3-69)*• Multiples of 3