Beginning Algebra, 11th Edition

The final result on the right (which is also valid for ) is called the

quadratic formula.It gives two values: one for thesign and one for thesign.

a 60

SECTION 9.3 Solving Quadratic Equations by the Quadratic Formula 569

Step 4 Use the square root property to complete the solution.

x

b 2 b^2  4 ac

2 a

x=

- 1  241

4

x=-

b

2 a



2 b^2 - 4 ac

2 a

x=-

1

4



241

4

x+

b

2 a

=

2 b^2 - 4 ac

2 a

x+

1

4

=

241

4

x+

b

2 a

=

B

b^2 - 4 ac

4 a^2

x+

1

4

=

B

41

16

Quadratic Formula

The solutions of the quadratic equation are

and

or, in compact form, x

b 2 b^2  4 ac

2 a

.

x

b 2 b^2  4 ac

2 a

x

b 2 b^2  4 ac

2 a

ax^2 +bx+c=0,aZ0,

Solving a Quadratic Equation by the Quadratic Formula

Solve

In this equation, and

Quadratic formula

Substitute

and

Simplify.

Add.

x= 2121 = 11

7  11

4

x=

7  2121

4

x=

7  249 + 72

4

b=-7, c=-9.

a=2,

x=

- 1 - 72  21 - 722 - 41221 - 92

2122

x=

• b 2 b^2 - 4 ac
  2 a

a=2,b=-7, c=-9.

2 x^2 - 7 x- 9 =0.

EXAMPLE 2

This represents

twosolutions.

Be sure to write in

the numerator.

• b

NOW TRY
EXERCISE 2
Solve 3x^2 + 5 x- 2 =0.

NOW TRY ANSWER

2. E-2,^13 F

Find the two solutions by first using the plus sign and then using the minus sign.

or

Checkeach solution. The solution set is E-1,^92 F. NOW TRY

x=

7 - 11

4

=

- 4

4

x= = - 1

7 + 11

4

=

18

4

=

9

2