Complete Square Two - Intermediate Algebra - Lecture Slides, Slides of Algebra

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§8.1 Complete

The Square

Review §

 Any QUESTIONS About

• §7.7 → Complex Numbers

 Any QUESTIONS About HomeWork

• §7.7 → HW-

7.7 MTH 55

SQUARE ROOT PROPERTY

• For any nonzero real number d , and any

algebraic expression u , then the Equation u^2 =

d has exactly two solutions:

If u^2 = d then u = d or u = − d

 Alternatively in a ShortHand Notation:

If u = d then u = ± d

2

Example  Use SqRt Property

• Solve 5 x^2 = 15. Give exact solutions and

approximations to three decimal places.

• SOLUTION (^5) x^2 = 15

x^2 = 3

x = 3 or x = − 3.

Isolating x^2 Using the Property of square roots

 The solutions are which round

to 1.732 and −1.732.

Or x = ± (^3) ShortHand Notation

Example  Use SqRt Property

• Solve x^2 +14 = 32
• SOLN (^) x^2 + 14 = 32

x^2 = 18 x = ± 18

x = ± 3 2

x = ± 9 ⋅ 2

Subtract 14 from both sides to isolate x^2 Use the square root property Simplify by factoring out a perfect square

Example  Use SqRt Property

• Solve ( x + 3) 2 = 7
• SOLN

Using the Property of square roots

 The solutions are The check

is left for us to do Later

Solving for x

( x + 3) 2 = 7

x + 3 = 7 or x + 3 = − 7

x = − 3 + 7 or x = − 3 − 7. Or x = − 3 ± 7 ShortHand Notation

Solving Quadratic Equations

• To solve equations in the form ax^2 = b, first isolate

x^2 by dividing both sides of the equation by a.

• Solve an equation in the form ax^2 + b = c

by using both the addition and multiplication

principles of equality to isolate x^2 before

using the square root principle

• In an equation in the form ( ax + b ) 2 = c , notice the

expression ax + b is squared. Use the square root

principle to eliminate the square.

Example  Use SqRt Property

• Solve (5 x − 3)^2 = 4
• SOLN

Add 3 to both sides and divide each side by 5, to isolate x.

Use the square root property

( 5 x^ −^3 )^2 =^4 5 x − 3 = ± 4

or

5 x − 3 = ± 2 2 3 5

x = ±^ +

2 3 5

x = +^2 5

x = −^ +

x = 1 1 5

or x =

Example  Complete the Sq

• Solve x^2 + 10 x + 4 = 0
• SOLN:

( x + 5) 2 = 21

x^2 + 10 x + 4 = 0

x^2 + 10 x = − 4

x^2 + 10 x + 25 = –4 + 25

x + 5 = 21 or x + 5 = − 21

Using the property of square roots

Factoring

Adding 25 to both sides.

 The solutions are The check

is left for Later

Solving Quadratic Equations by

Completing the Square

• Write the equation in the form 1 · x^2 + bx = c.
• Complete the square by adding ( b /2) 2 to both

sides.

• ( b /2) 2 is called the “Quadratic Supplement”
• Write the completed square in factored form.
• Use the square root property to eliminate the

square.

• Isolate the variable.
• Simplify as needed_._

Example  Complete the Sq

• Solve 2 x^2 − 10 x = 9
• SOLN:

Combine the fractions.

Factor.

Add to both sides and simplify the square root.

5 2

Use the square root principle.

4

43 4

25 4

18 2

(^5 2) = + =  

  

 (^) x −

4

43 2

x −^5 = ±

2

43 2

x =^5 ±

2

x =^5 ±^43

Example  Complete the Sq

• Solve by Completing the Square:

3 x^2 + 7 x +1 = 0

• SOLUTION: The coefficient of the x^2 term

must be 1. When it is not, multiply or divide

on both sides to find an equivalent eqn with

an x^2 coefficient of 1.

3 x^2 + 7 x + 1 = 0 (^13) ( ) (^13) 3 x^2 + 7 x + 1 = ⋅ 0 Divide Eqn by 3

Example  Complete the Sq

• Solve: 3 x^2 + 7 x +1 = 0
• SOLN:

(^7 37) or 7 37 6 6 6 6

x + = x + = −

7 2 37 6 36

 (^) x +  =  

(^7 37) or 7 37 6 6 6 6

x = − + x = − −

7 2 37 6 36

 (^) x +  =  

Square Root Property

Isolate x

Taking the Square Root of Both Sides

Example  Taipei 101 Tower

• The Taipei 101 tower in Taiwan is 1670 feet

tall. How long would it take an object to fall to

the ground from the top?

• Familiarize : A formula for

Gravity-Driven FreeFall with negligible air-drag

is s = 16 t^2

• where
  • s is the FreeFall Distance in feet
  • t is the FreeFall Time in seconds