The difference of two squares

The Maths Habit

  1. Factorise x2 + 7x + 10

    (x + 2)(x + 5)

  2. Factorise x2 − 3x − 28

    (x − 7)(x + 4)

  3. Factorise x2 − 11x + 24

    (x − 3)(x − 8)

Work these out.
What do you notice?

102 − 9292 − 8282 − 72

19, 17, 15: each one is the two numbers added

What is the square root?

  1. 49

    7

  2. 144

    12

  3. x2

    x

  4. 16x2

    4x

Expand and simplify

(x + 3)(x − 3) x2 − 9

x2 − 9: a square with a small square taken away

xx33

The small square is 3 by 3: area 9

Cut what is left into two rectangles

xx − 3x − 33

Pink: x by x − 3   Blue: x − 3 by 3

Turn the blue piece and stand it on the right

x + 3x − 3

x2 − 9 = (x + 3)(x − 3)

The quick way: square root each term.

Factorise x2 − 121

x11take the smallsquare awaysquare root each term
√ of x2 isx√ of 121 is11
one bracket with +, one with −
(x+11)(x−11)
Answer(x + 11)(x − 11)

Factorise x2 − 4

x2take the smallsquare awaysquare root each term
√ of x2 isx√ of 4 is2
one bracket with +, one with −
(x+2)(x−2)
Answer(x + 2)(x − 2)

Factorise x2 − 121

x11take the smallsquare awaysquare root each term
√ of x2 isx√ of 121 is11
one bracket with +, one with −
(x+11)(x−11)
Answer(x + 11)(x − 11)

Factorise y2 − 100

y10take the smallsquare awaysquare root each term
√ of y2 isy√ of 100 is10
one bracket with +, one with −
(y+10)(y−10)
Answer(y + 10)(y − 10)

Factorise x2 − 121

x11take the smallsquare awaysquare root each term
√ of x2 isx√ of 121 is11
one bracket with +, one with −
(x+11)(x−11)
Answer(x + 11)(x − 11)

Factorise x2 − 36

x6take the smallsquare awaysquare root each term
√ of x2 isx√ of 36 is6
one bracket with +, one with −
(x+6)(x−6)
Answer(x + 6)(x − 6)

Factorise x2 − 121

x11take the smallsquare awaysquare root each term
√ of x2 isx√ of 121 is11
one bracket with +, one with −
(x+11)(x−11)
Answer(x + 11)(x − 11)

Factorise a2 − 1

a1take the smallsquare awaysquare root each term
√ of a2 isa√ of 1 is1
one bracket with +, one with −
(a+1)(a−1)
Answer(a + 1)(a − 1)

Factorise x2 − 121

x11take the smallsquare awaysquare root each term
√ of x2 isx√ of 121 is11
one bracket with +, one with −
(x+11)(x−11)
Answer(x + 11)(x − 11)

Factorise m2 − 144

m12take the smallsquare awaysquare root each term
√ of m2 ism√ of 144 is12
one bracket with +, one with −
(m+12)(m−12)
Answer(m + 12)(m − 12)

The difference of two squares worksheet, page 1

Answer sheet

  1. Work out 512 − 492 without a calculator.

    200

  2. Factorise 4x2 − 9

    (2x + 3)(2x − 3)

  1. Factorise x2 − 64

    (x + 8)(x − 8)

  2. Factorise 16x2 − 81

    (4x + 9)(4x − 9)

  3. Use the difference of two squares to work out 632 − 372 without a calculator.

    2600
Exit tickets to print