Essential Question: What are some things the discriminate is used for?

5-8: The Quadratic FormulaSometimes you’ll run into a quadratic

expression that cannot be factored.For example: x2 + 10x + 4 = 0

There is no combination of numbers that multiplies to get 4 and adds to 10.

Yet, there are real numbers that exist for x to make that a true statement.

There exists a formula that allows you to find the solutions for any quadratic equation, called the QUADRATIC FORMULA

5-8: The Quadratic FormulaA quadratic equation written in standard

formax2 + bx + c = 0

can be solved with the quadratic equation

2 4

2

b b acx

a

5-8: The Quadratic FormulaThe “b2 – 4ac” underneath the square root is

called the discriminant.

The discriminant tells us how many (and what type) of solutions we get from the quadratic equation

2

2

4ab

a

bx

c

Discriminant

5-8: The Quadratic EquationThe Discriminant

Determine the type and number of solutions ofx2 + 6x + 8 = 0

a = 1, b = 6, c = 8Two real solutions

Value of discriminant

Type and Number of solutions

b2 – 4ac > 0 (positive)

2 Real Solutions

b2 – 4ac = 0 (zero) 1 Real Solution

b2 – 4ac < 0 (negative)

0 Real Solutions2 Imaginary Solutions

2

2

4

(6) 4(1)(8)

36 32

4

b ac

5-8: The Quadratic EquationDetermine the type and number of solutions

x2 + 6x + 9 = 0

x2 + 6x + 10 = 0

62 – 4(1)(9) = 36 – 36 = 01 Real Solution

62 – 4(1)(10) = 36 – 40 = -42 Imaginary Solutions

5-8: The Quadratic FormulaWhat is the quadratic equation?:

2 4

2

b b acx

a

5-8: The Quadratic FormulaEmpirical verification that the formula works

x2 + 8x + 12 = 0 can be factored as(x + 6)(x + 2) = 0 meaningx = -6 OR x = -2

a = 1, b = 8, c = 122

2

4

2

4(1)(1(8) (8

2

8

(1)

16

28 4

2

2))

b b acx

a

8 4 42

2 28 4 12

62 2

x

x

5-8: The Quadratic FormulaUsing the quadratic formula to solve a

problem that can’t be factoredx2 + 10x + 4 = 0

a = 1, b = 10, c = 4

2

2 (1)

(1

(4)

4

2

4

2

10 84

2

10 2 21 10 2 21

2 2

(10) (10)

2

5 21

)

b b acx

a

5-8: The Quadratic FormulaAnother equation that can’t be factored

2x2 - 6x + 1 = 0a = 2, b = -6, c = 1

2

2 (1)

4

2

4

2

(2)

(2

6 28

4

6 2 7 6 2 7

4 4 4

3

( 6) )

7

( 6

2

)

b b acx

a

5-8: The Quadratic FormulaUse the quadratic equation to solve 2x2 = –6x

– 7

2 4

2

b b acx

a

22 6 7 0x x , 72, 6 cba

2 4(( 26) )((6) 7)

22( )

6 36 56

4

6 20

4

6 20

4

i

6 4 5

4

i

6 2 5

4

i

3 5

2

i

5-8: The Quadratic FormulaAssignment

Page 293Problems 1 – 21, 31 – 39 (odds)

Show your work Ignore directions in 23-29 to approximate radical

solutions Leave your answers in simplest radical form