MCF3M Functions & Applications · Unit 2 — The Algebra of Quadratic Expressions
Lesson 2.5
Factoring Quadratic Expressions: Special Cases
Test special patterns, factor out a GCF first, and check your factors by expanding.
Estimated time: 30–45 minutes
Learning goals
- recognize and factor differences of squares
- recognize positive and negative perfect-square trinomials by checking 2AB
- distinguish special patterns from ordinary trinomials and sums of squares
- take out a GCF first and factor completely
- verify factors by expanding
Step 1 of 4
Warm Up
Success criteria
Step 2 of 4
Learn
Two patterns, one habit
First check for a greatest common factor (GCF). Then look at the terms left inside the brackets. A special pattern is a shortcut that comes from expanding — not a guess based on appearance.
Difference of squares
A difference means subtraction. A² − B² has two squared terms with a minus between them. Its factors are conjugate binomials: the same two terms, but opposite signs.
(A − B)(A + B) = A² + AB − AB − B²Multiply each term in one bracket by each term in the other.A² − B² = (A − B)(A + B)The opposite middle products cancel.
Worked example
Factor 36x² − 49.
Find the squares
Square the coefficient and variable together. Here A = 6x and B = 7.
(6x)² − 7²
Use opposite signs
A difference of squares needs conjugate brackets.
(6x − 7)(6x + 7)
Check by expanding
The +42x and −42x cancel.
36x² + 42x − 42x − 49 = 36x² − 49
Answer: (6x − 7)(6x + 7)
Perfect-square trinomials
A trinomial has three terms. Two matching brackets give a perfect-square trinomial. The two middle products match, so they add to 2AB. Check the middle term, not just the ends.
(A + B)(A + B) = A² + AB + AB + B²Two matching AB products add.A² + 2AB + B² = (A + B)²A positive middle uses matching plus signs.A² − 2AB + B² = (A − B)²Two negative AB products give a negative middle.
Worked example
Factor 4x² + 28x + 49.
Check the ends
The square roots of the ends are 2x and 7.
(2x)² + 28x + 7²
Test the middle
Calculate 2AB; it must equal the actual middle term.
2(2x)(7) = 28x
Factor
All three terms fit, and the middle is positive.
(2x + 7)²
Verify
Multiply the matching brackets.
4x² + 14x + 14x + 49 = 4x² + 28x + 49
Answer: (2x + 7)²
Worked example
Factor 9x² − 24x + 16.
Find A and B
A = 3x, B = 4; both ends are squares.
(3x)² − 24x + 4²
Check magnitude and sign
The middle is negative, with magnitude 2AB.
2(3x)(4) = 24x
Factor
Use two matching minus signs.
(3x − 4)²
Verify
The middle products are both −12x.
9x² − 12x − 12x + 16 = 9x² − 24x + 16
Answer: (3x − 4)²
Square ends are not enough
Worked example
Factor completely 5x² − 125.
GCF first
Both terms are divisible by 5.
5(x² − 25)
Keep going
Stopping here is incomplete: x² − 25 still factors.
5(x² − 5²)
Apply the difference pattern
Keep the outside factor 5.
5(x − 5)(x + 5)
Verify
Expand inside, then distribute 5.
5(x² − 25) = 5x² − 125
Answer: 5(x − 5)(x + 5)
A square display
A school display is a square. Its side is x + 6 cm, where x ≥ 0 is an adjustable length. Both dimensions match, so its area is side × side.
The mathematical model is A(x) = (x + 6)² = x² + 12x + 36. The input x is a length in centimetres; the output A(x) is area in square centimetres. This is a perfect-square trinomial. Factoring the area recovers the equal side lengths.
Worked example
Recover the side from area x² + 12x + 36 cm².
Test the pattern
The roots of the end terms are x and 6.
2(x)(6) = 12x
Factor the area
The brackets match because the display is square.
x² + 12x + 36 = (x + 6)(x + 6)
Interpret
For x ≥ 0 the side is positive.
Side = x + 6 cm
Answer: x + 6 cm
Algebra workspace · Pattern Detective
Test each expression, then choose its factors. For neither, read why the pattern fails.
Predict first
x² + 16 has square terms. Does that make it a difference of squares?
Make a prediction to open the workspace.
Step 3 of 4
Practice
Build it
Challenge
Explain it
Apply it
Step 4 of 4
Mastery Check
Lesson 2.5 mastery check
6 questions · no hints during the check. Answer them all, then check your score — you can retry as many times as you like.