10.7

10.7 Solve Quadratic Equations by the Quadratic Formula

Derive the Quadratic Formula

The Quadratic FormulaThe roots of the polynomial and

the solutions of the quadratic equation

are where and

Example 1 Find the roots of a polynomial

Find the roots of x2

– 6x + 3.

SOLUTION

The roots of are the values of x for whichx2

– 6x + 3x2

– 6x + 3 0.=

=x( )6–– 4

2( )1

–( )26– ( )1 ( )3 Substitute values in the quadraticformula: a 1, b 6, and c 3.= = – =

–+

=x2a

– b –b2 4acQuadratic formula

–+

Example 1 Find the roots of a polynomial

Simplify radical.2

=6 62–+

Divide out factor of 2.=2

3 6( )2= –+3 6–+

Simplify.2

6=

24–+

ANSWER

The roots of arex2

– 6x + 3 +3 6 –3 6.and

Example 1 Find the roots of a polynomial

CHECK Substitute each root for x. The polynomial should simplify to 0.

+3 6( )2 +3 6( )– 6 + 3

= 9 + 66 + 6 – 18 – 66 + 3

–3 6( )2 –3 6( )– 6 + 3

= 9 – 66 + 6 – 18 + 66 + 3

= 0

= 0

Multiple Choice PracticeExample 2

SOLUTION

Write original equation.2x2 – 7 = x

2x2 – x = 0– 7Write in standard form.

Which is one of the solutions to the equation2x2 – 7 = x?

4

157–

4

157+

4

+1 57–

4

+1 57

Multiple Choice PracticeExample 2

ANSWER

One solution is

The correct answer is D.4

+1 57.

Quadratic formula=x

2a

– b –b2 4ac–+

Simplify.4

1 57=

–+

( )1–– – 4( )21– ( )2 ( )7

2( )2

–=

–+Substitute values in the quadratic formula: a 2, b –1, and c –7.

= ==

Methods for Solving Quadratic Equations

Example 4 Choose a solution method

Tell what method(s) you would use to solve the quadratic equation. Explain your choice(s).

10x2 – 7 = 0a.

x2 + 4x = 0b.

+ 9x5x2 – 4 = 0c.

SOLUTION

a. The quadratic equation can be solved using square roots because the equation can be written in the form x2 d. =

Example 4

b.

Choose a solution method

The quadratic equation can be solved by factoring because the expression x2 4x can be factored easily. Also, the equation can be solved by completing the square because the equation is of the form where a 1 and b is an even number.

+

ax2 + bx + c = 0 =

c. The quadratic equation cannot be factored easily, and completing the square will result in many fractions. So, the equation should be solved using the quadratic formula.

Example 3 Use the quadratic formula

For the period 1971 2001, the number y of films produced in the world can be modeled by the function where x is the number of years since 1971. In what year were 4200 films produced?

FILM PRODUCTION

–

+ 3900= 10x2 – 94xy

SOLUTION

+ 3900= 10x2 – 94xy Write function.

+ 3900= 10x2 – 94x4200 Substitute 4200 for y.

300= 10x2 – 94x0 Write in standard form.–

Example 3 Use the quadratic formula

Substitute values in the quadratic formula: a 10, b –94, and c –300.

===

– –+( )94– – 4( )294– ( )10 ( )300

2( )10

–=x

ANSWER

There were 4200 films produced about 12 years after 1971, or in 1983.

20

–+94 20,836= Simplify.

The solutions of the equation are20

+94 20,836≈ 12 and

20

–94 20,836≈ –3.

10.7 Warm-up (Day 1)Use the quadratic formula to Find the roots

1.

Use the quadratic formula to solve the equation

2.

3.

4.

10.7 Warm-up (Day 2)Use the quadratic formula to solve the equation

1.

2.

10.7 Warm-up (Day 3)Use the quadratic formula to solve the equation

1.

2.