Difference of Two Squares and Perfect Square Trinomials

Mathematics · Algebra · ages 14-15

Name ______________________ Date ____________

x squared + 36 factorises as (x + 6)(x - 6).
Circle one: True False
Factorise 16x squared - 9.
- (16x + 9)(16x - 9)
- (4x + 3)(4x - 3)
- (4x + 9)(4x - 3)
Factorise x² - 144.
- (x + 12)(x + 12)
- (x + 12)(x - 12)
- (x - 12)(x - 12)
- (x + 72)(x - 2)
Factorise x squared - 36.
- (x + 6)(x - 6)
- (x + 6)(x + 6)
- (x - 6)(x - 6)
What is (x + 10)² fully expanded?
- x² + 100
- x² + 10x + 100
- x² + 20x + 20
- x² + 20x + 100
Which shows x² + 18x + 81 fully factorised?
- (x + 9)(x - 9)
- (x + 9)²
- (x - 9)²
- (x + 18)(x + 18)
Factorise x squared + 16x + 64.
- (x + 16) squared
- (x + 8) squared
- (x + 4) squared
Leo factorises x squared - 16 as (x - 4) squared. What is his error?
- (x - 4) squared expands to x squared - 8x + 16, so a middle term survives
- 16 is not a perfect square
- He should have written (x + 4) squared
Factorise 4x squared + 12x + 9.
- (4x + 3) squared
- (2x + 9) squared
- (2x + 3) squared
Use a difference of two squares to work out 98 times 102.
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

False · The two squares are joined by a plus, so the difference of two squares pattern does not apply.
(4x + 3)(4x - 3) · 16x squared is (4x) squared and 9 is 3 squared, so the brackets are (4x + 3)(4x - 3).
(x + 12)(x - 12) · 144 is 12 squared, so x² - 144 splits as (x + 12)(x - 12).
(x + 6)(x - 6) · 36 is 6 squared, so the difference of two squares gives (x + 6)(x - 6).
x² + 20x + 100 · Squaring doubles the middle: 2 times 10x gives 20x, and 10 squared is 100.
(x + 9)² · Half of 18 is 9 and 9 squared is 81, so it folds to (x + 9)².
(x + 8) squared · Halve the middle: 16 over 2 gives 8, and 8 squared is 64, so it is (x + 8) squared.
(x - 4) squared expands to x squared - 8x + 16, so a middle term survives · A squared bracket keeps a middle term. x squared - 16 is a difference of two squares, so it needs (x + 4)(x - 4).
(2x + 3) squared · The middle of (2x + 3) squared is 2 times 2x times 3 = 12x, and 3 squared is 9, so it fits.
9996 · 98 times 102 = (100 - 2)(100 + 2) = 100 squared - 2 squared = 10000 - 4 = 9996.