Perfect square

COMPLETING THE SQUARE

Do you remember?Do you remember?

(x + 2)2 = x2 + 4x + 4

and

(x – 2)2 = x2 – 4x + 4

Remember the SQUARE ROOT PROPERTY?

Remember the SQUARE ROOT PROPERTY?

• For any real number n,

if x2 = n, then x = + n

• For any real number n,

if x2 = n, then x = + n

For example:

x2 = 16 therefore x = + 4

x2 = 5 therefore x = + 5

Remember: PERFECT SQUARE TRINOMIALS?


• x2 + 8x + 16• x2 + 8x + 16

The first term must be a perfect square…

x2 = ( x )( x )



x2 + 8x + 16 x2 + 8x + 16

The third term must be a perfect square…

16 = ( 4 )( 4 )



x2 + 8x + 16 x2 + 8x + 16

The middle term must

equal the sum of the

factors of the third term.

4 + 4 = 8



Now factor

x2 + 8x + 16 =

Now factor

x2 + 8x + 16 =

( x + 4 )( x +4 ) or ( x +4)2



x2 + 4x + 4 x2 + 4x + 4

( x )( x ) + 2x + 2x + (2)(2)

Perfect squares

Add the middle term.

OKAY…ARE WE READY TO SOLVE Quadratic Equations

using…

OKAY…ARE WE READY TO SOLVE Quadratic Equations

using…

Equation of Rational RootsEquation of Rational Roots

Problem: x2 + 10x + 25 = 49Problem: x2 + 10x + 25 = 49

Step 1: x2 + 10x + 25 = 49

NOTICE: This is a PERFECT SQUARE TRINOMIAL

FACTOR

Step 2: Use the Square Root Property on your Factored Trinomial.

Step 2: Use the Square Root Property on your Factored Trinomial. ( x + 5 )2 = 49

x + 5 = + 49

( x + 5 )2 = 49

x + 5 = + 49

Step 3: Remember + 49 = 7

x + 5 = + 7Therefore:

Take the square of both sides!

Step 4: Subtract 5 from both sides.Step 4: Subtract 5 from both sides.

x + 5 = + 7 Therefore

x = - 5 + 7 Or

x = - 5 - 7

x + 5 = + 7 Therefore

x = - 5 + 7 Or

x = - 5 - 7

Step 5: Simplify.Step 5: Simplify. x = - 5 + 7 x = -5 -7 x = 2 or x = -12

Therefore the solution of

X2 + 10x + 25 = 49

is

{-12, 2}

x = - 5 + 7 x = -5 -7 x = 2 or x = -12

Therefore the solution of

X2 + 10x + 25 = 49

is

{-12, 2}

Now It’s YOUR TURN !!!!Now It’s YOUR TURN !!!!

Solve:Solve:

x2 + 14x + 49 = 64x2 + 14x + 49 = 64

And the answer is

{-15, 1}

Here’s how…

x2 + 14x + 49= 64(x + 7)2 = 64 x + 7 = 64 x + 7 = + 8 x = -7+ 8 or x = -7 - 8 x = 1 or x = -15 so {-15, 1}

Equations with Irrational Roots!

Equations with Irrational Roots!

And the problem is…

x2 – 6x + 9 = 32

Step 1: Factor the perfect squareStep 1: Factor the perfect square

x2 - 6x + 9 = 32(x - 3)2 = 32 x2 - 6x + 9 = 32(x - 3)2 = 32

Step 2: Square Root Property

x - 3 = + 32

Step 3: Factor the perfect square

Also remember 32 = 4 2

Step 3: Factor the perfect square

Also remember 32 = 4 2

Step 4: Write as 2 equations.

x = 3 + 4 2 or x = 3- 4 2

x = 3 + 4 2

Now It’s YOUR TURN !!!!Now It’s YOUR TURN !!!!

SolveSolve

x2 - 10x + 25 = 12x2 - 10x + 25 = 12

And the answer is

{5 + 2 3 }

Here’s how…

x2 - 10x + 25= 12(x - 5)2 = 12 x - 5 = 12 x - 5 = + 2 3 x = 5 + 2 3

{x = 5 + 2 3}

Assessment LinksAssessment Links

http://www.mccc.edu/~kelld/quadratic/quadratic.htm

Click on address to hyperlink to the internet

A. 18 Practice questions

B. Practice Quiz

http://www.glencoe.com/sec/math/algebra/algebra2/algebra2_03/

Education

Perfect square