Quadratic Formula and the Discriminant

Essential Questions

How do I use the QUADRATIC FORMULA to solve equations?

How do I use the DISCRIMINANT to determine the # of solutions and the factorability for a quadratic equation?

Warm Up

Evaluate each expression. 1) 62 – 4(1)(3) 2) 22 – 4(1)(-3) 3)

Answers: 24, 16, -1

)1(2

)5)(1(466 2

I will learn to…

10.5 The Quadratic Formula

Use the quadratic formula to find solutions to quadratic equations.

Evaluate the discriminant to determine how many real roots a quadratic equation has (how many solutions there will be).

Vocabulary

10.5 The Quadratic Formula

Quadratic Formula: formula used to solve a quadratic equation. Always works! Normally used when you CANNOT factor.

Discriminant: expression from the quadratic formula that helps you determine how many real solutions there will be in the quadratic equation. If D < 0 : No Real SolutionsIf D = 0: Exactly One Real SolutionIf D > 0: Two Real Solutions

Rules and Properties

The Quadratic Formula

10.5 The Quadratic Formula

x = –b b2 – 4ac

2a

For ax2 + bx + c = 0, where a 0:

The Discriminant: b2 – 4ac

Example 1

Solve using the quadratic formula.2x2 + 5x + 1 = 0a = 2, b = 5, c = 1Plug in!!

25 5 4(2)(1)

2(2)x

x = –b b2 – 4ac

2a

5 17

4x

Example 2 Solve:3x2 + 2x – 4 = 0

Can you factor it? NO Use the quadratic formula! A: 3 B: 2 C: -4

Example 2 Continued A: 3 B: 2 C: -4

22 2 4(3)( 4)

2(3)x

2 4 ( 48)

6x

2 52

6x

2 2 13

6x

1 13

3x

Example 3Solve using Quad. Formula

x2 + 5x – 8 = 0 (this can’t factor!!)a = 1, b = 5, c = –8

x = –5 52 – 4(1)(–8)

2(1)

–5 57

2=

You Try

Solve using the quadratic formula.3x2 – 4x – 2 = 0 a = 3, b = -4, c = -2

“You Try” Continueda = 3, b = -4, c = -2

24 ( 4) 4(3)( 2)

2(3)x

4 16 24

6x

4 40

6x

4 2 10

6x

2 10

3x

Ex. 4) Find the # of real solutions.

Remember use the discriminant, b2 – 4ac. 3x2 – 2x + 1 = 0 (-2)2 – 4(3)(1) 4 – 12 -8 There are NO real solutions! By finding the discriminant FIRST ALWAYS, you can save

yourself some time if its not workable! And if you DO discover there are some solutions, you have

just already completed part of the Q.F.! So, you didn’t waste time here either.

More with a +Discriminant

If the discriminant is a perfect square then the equation is factorable!

Don’t use the quadratic formula if you can factor…it is a waste of time.

Ex. If D = 49 you can factor to solve. Ex. If D = 48 you can only use Q.F.

Think About It

What part of the Q.F. is the discriminant?

The part under the radical!!