a

acbbx

2

42

Getting Started

The objective is to be able to solve any quadratic equation by using the quadratic formula.

Quadratic Equation - An equation in x that can be written in the standard form ax2 + bx + c = 0, a 0.

Discriminant - The expression b2 - 4ac, for a quadratic equation ax2 + bx + c = 0.

How Many Solutions Will There Be?

To determine how many solutions the quadratic equation will have, determine the value of the discriminant.

If b2 - 4ac is positive, then the equation has two solutions.

If b2 - 4ac is zero, then the equation has one solution.

If b2 - 4ac is negative, then the equation has no solution.

How Many Solutions Are There?

How many solutions do each of the following equations have? 232 .1 2 xx

25169

)2)(2(434 22 acb

Since b2 - 4ac is positive, there will be two solutions to the equation.

52 .2 2 xx

4 20

16

b ac2 24 2 4 1 5 ( ) ( )( )

Since b2 –4ac is negative, there will be no solution to the equation.

Use the quadratic formula to solve the following equation.

x x2 6 7 0

ax

b b ac 2 42

x 6 36 282

x 6 82

x 26 6 4 1 72 1

( ) ( ) ( )( )( )

x6 8

2

2

x 6 4 2

2

2x 6 2 2

2x 3 2

x 6 8

2

x 6

2

4 2

2

2x 6 2 2

2

x 3 2

a b c 1 6 7, ,

x x 4 41 1 59. . o r

Determine how many solutions each equation has by using the discriminant.

1 2 4 2 02. x x

2 2 4 02. x x

31

23 02. x x

Use the quadratic formula to solve the equation.

4 5 2 2 02. y y 5 2 4 3 02. x x

2 4 2 02x x

a b c 2 4 2

16 16

0

b ac2 24 4 4 2 2 ( )( )

16 4 4 ( )

Since the value of the discriminant is zero, there is only one solution to this equation.

x x2 2 4 0

a b c 1 2 4

4 16

12

b ac2 24 2 4 1 4 ( ) ( )( )4 4 4 ( )

Since the value of the discriminant is negative, there is no solution to this equation.

1

23 02x x

a b c 1

21 3

2 b ac2 24 1 4

13

1 6

7

1 2 3 ( )

Because the value of the discriminant is positive, this equation will have two solutions.

5 2 2 02y y

a b c 5 2 2

2 4 40

10

10 2 44

10

5

1 4 11

10

1

5

2 11

10

b b ac

a

2 24

2

2 2 4 5 2

2 5

( ) ( ) ( )( )

( ) 1 11y

5y 0 86.

y 0 46.

or

2 4 3 02x x

a b c 2 4 3

4 404

4 4 10

4104 2

4

4 4 16 24

222a

b b ac2 24 4 4 4 2 3( )( )( ) 1

10

20 58.

2 58.

or