Seia2e_1001 10.1 Solving Quadratic Equations by Completing the Square

8/2/2019 Seia2e_1001 10.1 Solving Quadratic Equations by Completing the Square

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Chapter 10

QuadraticEquations and

Functions

Section 1

Solving QuadraticEquations by

Completing the Square


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Sullivan, III & Struve,Elementary and Intermediate Algebra 10.1 - 2Copyright 2010 Pearson Education, Inc.

Section 10.1 Objectives

1 Solve Quadratic Equations Using the Square

Root Property

2 Complete the Square in One Variable

3 Solve Quadratic Equations by Completing the

Square

4 Solve Problems Using the Pythagorean

Theorem


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Square Root Property

236x


Ifx2 =p, thenx = orx =p .p

Example: Solve the equation:

36x

6x


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Solving Quadratic Equations Using the Square

Root Property

Step 1: Isolate the expression containing the square

term.Step 2: Use the Square Root Method. Dont forget

the symbol.

Step 3: Isolate the variable, if necessary.

Step 4: Check. Verify your solutions.


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Example: Solve the equation:2

3 150x 2

50x

50x

5 2x

5 2x Check: 5 2x 2

3( )5 2 150

3(25 2) 150

3(50) 150

150 150

23 5 2( ) 150

3(25 2) 150

3(50) 150

150 150

Divide each side by 3.

Take the square root of each side.

Simplify.


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Example: Solve:2

(2 3) 10x

2(2 3) 10x Take the square root of each side.

2 3 10x Simplify.

2 10 3x Subtract 3 from both sides.

10 3

2x

Divide both sides by 2.

Continued.


7/17Sullivan, III & Struve,Elementary and Intermediate Algebra 10.1 - 7Copyright 2010 Pearson Education, Inc.


2(2 3) 10x

Example continued:10 32

x Check:

10 32

x

2

10 310

22 3

10 3 3

2

10

10 2 10

10 10

2(2 3) 10x

2

10 3

22 3 10

10 3 3

2

10

10 2 10

10 10




The idea behind completing the squareis to adjust the leftside of a quadratic equation of the formx2 + bx + c in order to

make it a perfect square trinomial.

Obtaining a Perfect Square TrinomialStep 1: Identify the coefficient of the first-degree term.

Step 2: Multiply this coefficient by and then square the result.

Step 3: Add this result to both sides of the equation.

1

2

26 5x x

coefficient of the

first-degree term 2

16 9

2

25 96 9x x


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Example: Solve: x2

6x

7 = 0

Add 7 to both sides.x26x = 7

Continued.

Complete the square in the

expressionx26x.

x26x + 9= 7 + 9

2

16

2

2( 3) 9

Factor the left side.(x23) = 16

Use the Square Root Property.3 16x Simplify.3 4x

Add 3 to both sides.3 4x

7 or 1x x




7 or 1x x Example continued:

Check: x26x7 = 0

726(7)

7 = 0

49427 = 0

0 = 0

x26x7 = 0

(1)26(1)

7 = 0

1 + 67 = 0

0 = 0




Example: Solve: 2x2

+ 5x3 = 0

Divide each term by 2.

2x2 + 5x = 3

2 5 3

2 2

x x

2 5 25 3 25

2 16 2 16x x Complete the square in the

expression 2 5 .2

x x

Continued.

Add 3 to both sides.

21 5

2 2

5

4

2 25

16

x 5

4

2

49

16Factor the left side.




22 5 3 0x x

Example continued:

Check:

Use the Square Root Property.5 494 16

x

Simplify.5 7

4 4x

7 5

4 4x Subtract from both sides.

5

4

12 2 13 or

4 4 2x x

22( ) 5( )3 3 03

2(9) 15 3 0

0 0

22 5 3 0x x

2 12

2

5 12

3 0

21

4

52

3 0

1 5 60

2 2 2



Pythagorean Theorem

Pythagorean Theorem

In a right triangle, the square of the length of the

hypotenuse is equal to the sum of the squares of the

lengths of the legs.

c2 = a2 + b2.

a

bchypotenuse

legs



Pythagorean Theorem

Example:A baseball diamond is square. Each side of the square is 90 feet

long. How far is it from home plate to second base?

Step 1: Identify We want to know how far it is from home plate

to second base. Homeplate

Second

base

Let c be the distance

from home plate tosecond base.

c

Continued.

Step 2: Name

90

90



Pythagorean Theorem

Example continued:

c2 = a2 + b2

c2 = 902 + 902 Substitute.

Use the Pythagorean Theorem.

Continued.

Step 3: Translate

Step 4: Solve c2 = 8100 + 8100

c2 = 16200

16200c

127.3

90

90

c


17/17Sullivan III & Struve Elementary and Intermediate Algebra 10 1 17Copyright 2010 Pearson Education Inc

Pythagorean Theorem

127.3c

Example continued:

Step 5: Check

127.3 is not used because length is never negative.

c2 = a2 + b2

127.32 = 902 + 902

16205.29 = 8100 + 8100

16205.29

16200

Due to rounding error.

Step 6: Answer The distance from home plate to second base is

approximately 127.3 feet.

Seia2e_1001 10.1 Solving Quadratic Equations by Completing the Square

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