MPM2D
Unit 4 – Lesson #4
4.5 – Factoring Quadratics: SPECIAL CASES
GOAL: Factor perfect-square trinomials and difference of squares.
Special Case #1: DIFFERENCE OF SQUARES
Discovery: Consider the expansion of the following:

o
(x + 3)(x – 3) =
o
(x – 4)(x + 4) =
Therefore, a polynomial of the form a2 – b2 is a difference of squares and can be factored
as ______________________________.

In general:
o A difference of squares expression always has exactly ______ terms.
o
Each of the terms is a _______________ number.
o
The terms are separated by a ____________ sign.
Steps for Factoring Difference of Squares
Example: 4x2 – 9
1.) Check:
Are both terms perfect squares?
Are the terms separated by a minus sign?
2.) Find the square root of each term.
(Find the square root of a2, which will be “a”
and the square root of b2, which will be “b”)
3.) Place the a and b values in the factored
form: (a + b)(a – b).
(Add the roots in one bracket, and
subtract the roots in another bracket).
Example #1: Factor the expressions.
a.) x2 – 25
b.) 9x2 – 49y2
c.) x4 - 16
d.) 8x2 – 50y2
MPM2D
Unit 4 – Lesson #4
Special Case #2: PERFECT-SQUARE TRINOMIALS
Discovery: Consider the expansion of:
o
(x + 3)(x + 3) =
o
(2x – 4)(2x – 4) =

What do you notice about the first term when expanded? _______________________________

What do you notice about the sign of the middle term when expanded? ____________________

What do you notice about the middle term when expanded? ____________________________

What do you notice about the third term when expanded? ______________________________

Therefore,
a2 + 2ab + b2 = ___________________________________
a2 – 2ab + b2 = ___________________________________

In general:
o
A perfect square trinomial always has a third term that is _____________.
o
The first and last terms are ________________ values.
o
The middle term is a value that is TWICE THE PRODUCT of the ________ ________ of
the first and last terms.
o
The middle term is _____________ if the factors have NEGATIVE signs in the brackets.
The middle term is ______________if the factors have POSITIVE signs in the brackets.
25x 2 + 70x + 49
Example #2: Determine whether or not the following expressions are perfect square trinomials.
a.) 16x2 - 80x + 100
b.) 25x2 + 12x - 30
c.) 36x2 + 54x + 81
MPM2D
Unit 4 – Lesson #4
Steps for Factoring Perfect Square Trinomials
1.) Check:
- Are the first and last terms perfect squares?
- Is the middle term twice the product of the
square roots of the first and last terms?
- Is the last term positive?
Example: 4x2 + 12x + 9
2.) Find the square root of the ________ term
and the square root of the ________ term.
First term = a2, Square Root = a
Last term = b2, Square Root = b
3.) Write the terms in a squared bracket (
)2,
separated by the sign of the middle term.
*Note: Perfect square trinomials can also be factored using the simple or complex trinomial
method. The two resulting brackets will be IDENTICAL. Always write the final answer as a single
bracket, all squared.
Ex: if you end up with (x - 4)(x - 4), you write it as _________________________
Example #3: Factor the expressions.
b.) 16x2 – 24x + 9
a.) 25x2 + 20x + 4
Example #4: Given the quadratic relation y = 12x2 – 60x + 75,
a) Express the relation in factored form.
e) Sketch the graph
b) Determine the zeros
c) Determine the y-intercept
.
d.) Determine the vertex.
Homework: p. 230-231, # 5, 6, 7abc, 9, 10abcd, 11af, 14a