perfect square
DESCRIPTION
Here is a powerpoint on identifing perfect squares, which we will need when looking at completing the squares.TRANSCRIPT
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?
Remember: PERFECT SQUARE TRINOMIALS?
• x2 + 8x + 16• x2 + 8x + 16
The first term must be a perfect square…
x2 = ( x )( x )
Remember: PERFECT SQUARE TRINOMIALS?
Remember: PERFECT SQUARE TRINOMIALS?
x2 + 8x + 16 x2 + 8x + 16
The third term must be a perfect square…
16 = ( 4 )( 4 )
Remember: PERFECT SQUARE TRINOMIALS?
Remember: PERFECT SQUARE TRINOMIALS?
x2 + 8x + 16 x2 + 8x + 16
The middle term must
equal the sum of the
factors of the third term.
4 + 4 = 8
Remember: PERFECT SQUARE TRINOMIALS?
Remember: PERFECT SQUARE TRINOMIALS?
Now factor
x2 + 8x + 16 =
Now factor
x2 + 8x + 16 =
( x + 4 )( x +4 ) or ( x +4)2
Remember: PERFECT SQUARE TRINOMIALS?
Remember: PERFECT SQUARE TRINOMIALS?
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/