Copyright © 2011 Pearson, Inc.

P.5Solving

Equations Graphically,

Numerically and Algebraically

Slide P.5 - 2 Copyright © 2011 Pearson, Inc.

What you’ll learn about

Solving Equations Graphically Solving Quadratic Equations Approximating Solutions of Equations Graphically Approximating Solutions of Equations Numerically

with Tables Solving Equations by Finding Intersections

… and whyThese basic techniques are involved in using a graphing

utility to solve equations in this textbook.

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Example Solving by Finding x-Intercepts

2Solve the equation 2 3 2 0 graphically.x x− − =

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Solution

2Find the -intercepts of 2 3 2.

Use the Trace to see that ( 0.5,0) and (2,0) are -intercepts.

Thus the solutions are 0.5 and 2.

x y x x

x

x x

= − −−

=− =

2Solve the equation 2 3 2 0 graphically.x x− − =

Slide P.5 - 5 Copyright © 2011 Pearson, Inc.

Zero Factor Property

Let a and b be real numbers.

If ab = 0, then a = 0 or b = 0.

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Quadratic Equation in x

A quadratic equation in x is one that can be written in the form

ax2 + bx + c = 0,

where a, b, and c are real numbers with a ≠ 0.

Slide P.5 - 7 Copyright © 2011 Pearson, Inc.

Completing the Square

To solve x 2 +bx=c by completing the square, add (b/ 2)2 toboth sides of the equation and factor the left side of the new equation.

x2 +bx+b2

⎛

⎝⎜⎞

⎠⎟

2

=c+b2

⎛

⎝⎜⎞

⎠⎟

2

x+b2

⎛

⎝⎜⎞

⎠⎟

2

=c+b2

4

Slide P.5 - 8 Copyright © 2011 Pearson, Inc.

Quadratic Formula

The solutions of the quadratic equation ax 2 +bx+c=0, where a≠0, are given by the quadratic formula

x=−b± b2 −4ac

2a.

Slide P.5 - 9 Copyright © 2011 Pearson, Inc.

Example Solving Using the Quadratic Formula

2Solve the equation 2 3 5 0.x x+ − =

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Solution

a =2, b=3, c=−5

x=−b± b2 −4ac

2a=−3± 32 −4 2( ) −5( )

2 2( )

=−3± 49

4=−3±74

x=−52 or x=1.

2Solve the equation 2 3 5 0.x x+ − =

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Solving Quadratic Equations Algebraically

These are four basic ways to solve quadratic equations algebraically.

1. Factoring

2. Extracting Square Roots

3. Completing the Square

4. Using the Quadratic Formula

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Agreement about Approximate Solutions

For applications, round to a value that is reasonable for the context of the problem. For all others round to two decimal places unless directed otherwise.

Slide P.5 - 13 Copyright © 2011 Pearson, Inc.

Example Solving by Finding Intersections

Solve the equation −2 x−2 =−3.

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Solution

Graph y =−2 x−2 and y=−3.

Use Trace or the intersectfeature of your grapher tofind the points of intersection.The graph indicates that thesolutions arex=0.5 and x=3.5.

Solve the equation −2 x−2 =−3.

Slide P.5 - 15 Copyright © 2011 Pearson, Inc.

Quick Review

( )( )( )

2

3 2

4 2

Expand the product.

1. 2

2. 2 1 4 3

Factor completely.

3. 2 2

4. 5 36

5. Combine the fractions and reduce the resulting fraction

2to lowest terms.

2 1 1

x y

x x

x x x

y y

x

x x

+

+ −

+ − −

+ −

−+ −

Slide P.5 - 16 Copyright © 2011 Pearson, Inc.

Quick Review Solutions

( )( )( )

( )( )( )( )( )( )

2 2

3 2

4 2 2

2

2

4 4

8 2 3

1

Expand the product.

1. 2

2. 2 1 4 3

Factor completely.

3. 2 2

4. 5 36

5. Combine the fractions and reduce the resulting fraction

to low

2

9 2

e

1

2

x xy y

x x

x x x

y y

x y

x x

x x

y y

x

y

+

+ −

+ − −

+

+ +

− −

+ − +

+ − +−

( )( )

22st terms.

2 1 11

5 2

2 1

xx

x x

x

x x−

+ −

− +

+ −