Here is a powerpoint on identifing perfect squares, which we will need when looking at completing the squares.

1. COMPLETING THE SQUARE

2. Do you remember?
(x + 2) = x + 4x + 4
2 2
and
(x – 2) = x – 4x + 4
2 2

3. Remember the
SQUARE ROOT PROPERTY?
• For any real number n,
if x2 = n, then x = + n
For example:
x2 = 16 therefore x = + 4
x2 = 5 therefore x = + 5

4. Remember: PERFECT SQUARE
TRINOMIALS?
• x2 + 8x + 16
The first term must be a
perfect square…
x = ( x )( x )
2

5. Remember: PERFECT SQUARE
TRINOMIALS?
x2 + 8x + 16
The third term must be a
perfect square…
16 = ( 4 )( 4 )

6. Remember: PERFECT SQUARE
TRINOMIALS?
x + 8x + 16
2
The middle term must
equal the sum of the
factors of the third term.
4+4=8

7. Remember: PERFECT SQUARE
TRINOMIALS?
Now factor
x2 + 8x + 16 =
( x + 4 )( x +4 ) or ( x +4)2

8. Remember: PERFECT SQUARE
TRINOMIALS?
x2 + 4x + 4
( x )( x ) + 2x + 2x + (2)(2)
Add the middle term.
Perfect squares

9. OKAY…ARE WE READY TO
SOLVE Quadratic Equations
using…

10. Equation of Rational Roots
Problem: x2 + 10x + 25 = 49
Step 1: x2 + 10x + 25 = 49
FACTOR
NOTICE: This is a PERFECT
SQUARE TRINOMIAL

11. Step 2: Use the Square Root
Property on your Factored Trinomial.
Take the square of both
( x + 5 ) = 49
2
sides!
x + 5 = + 49
Step 3: Remember + 49 = 7
Therefore:
x+5= +7

12. Step 4: Subtract 5 from both sides.
x+5 = +7
Therefore
x=-5+7
Or
x=-5-7

13. 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}

14. Now It’s YOUR TURN !!!!

15. Solve:
x + 14x + 49 = 64
2

16. And the answer
is
{-15, 1}

17. 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}

18. Equations with Irrational
Roots!
And the problem is…
x2 – 6x + 9 = 32

19. Step 1: Factor the perfect square
x - 6x + 9 = 32
2
(x - 3)2 = 32
Step 2: Square Root Property
x - 3 = + 32

20. Step 3: Factor the perfect square
Also remember 32 = 4 2
x= 3 + 4 2
Step 4: Write as 2 equations.
x = 3 + 4 2 or x = 3- 4 2

21. Now It’s YOUR TURN !!!!

22. Solve
x - 10x + 25 = 12
2

23. And the answer
is
{5 + 2 3 }

24. 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}

25. Assessment Links
Click on address to hyperlink to the internet
A. 18 Practice questions
http://www.mccc.edu/~kelld/quadratic/quadratic.htm
B. Practice Quiz
http://www.glencoe.com/sec/math/algebra/algebra2/algebra2_03/