Notes Packet 10: Solving Quadratic Equations by the

Quadratic Formula

THE QUADRATIC FORMULA

1. The general quadratic equation

2. This is the quadratic formula!

3. Just identify a, b, and c then substitute into the formula.

2 4

2

b b acx

a

2 0ax bx c

Identifying a, b, and c

From the quadratic equation,

ax2+bx+c=0The quadratic term is aThe linear term is bThe constant is c

Example:x2 + 2x – 5 = 0

a = 1b = 2c = -5

WHY USE THE QUADRATIC FORMULA?

The quadratic formula allows you to

solve ANY quadratic equation, even if

you cannot graph it.

An important piece of the quadratic

formula is what’s under the radical:

b2 – 4ac

This piece is called the discriminant.

WHY IS THE DISCRIMINANT IMPORTANT?

The discriminant tells you the number and types of

answers

(roots) you will get. The discriminant can be +, –, or 0

which actually tells you a lot! Since the discriminant is

under a radical, think about what it means if you have a

positive or negative number or 0 under the radical.

WHAT THE DISCRIMINANT TELLS YOU!

Value of the Discriminant

Nature of the Solutions

Negative 2 imaginary solutions

Zero 1 Real Solution

Positive – perfect square

2 Reals- Rational

Positive – non-perfect square

2 Reals- Irrational

Example #1

22 7 11 0x x

Find the value of the discriminant and describe the nature of the roots (real,imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula)

1.

a=2, b=7, c=-11

Discriminant = 2

2

4

(7) 4(2)( 11)

49

137

88

b ac

Discriminant =

Value of discriminant=137

Positive-NON perfect square

Nature of the Roots – 2 Reals - Irrational

Example #1- continued

22 7 11 0x x

2

2

4

2

7 7 4(2)( 11)

2(

2, 7, 11

7 137 2 Reals - Irrational

4

2)

a b

b ac

a

c

b

Solve using the Quadratic Formula

Solving Quadratic Equations by the Quadratic Formula

2

2

2

2

2

1. 2 63 0

2. 8 84 0

3. 5 24 0

4. 7 13 0

5. 3 5 6 0

x x

x x

x x

x x

x x

Try the following examples. Do your work on your paper and then check your answers.

1. 9,7

2.(6, 14)

3. 3,8

7 34.

2

5 475.

6

i

i

Complex Numbers

Definition of pure imaginary numbers:

Any positive real number b,

where i is the imaginary unit and bi is called the pure imaginary number.

Definition of pure imaginary numbers:

i is not a variable it is a symbol for a specific

number

Simplify each expression.

Remember

Simplify each expression.

Remember

Simplify.To figure out where we are in the cycle divide the exponent by 4

and look at the remainder.

Simplify.Divide the exponent by 4 and look at the remainder.

Simplify.Divide the exponent by 4 and look at the remainder.

Simplify.Divide the exponent by 4 and look at the remainder.