The Difference of Two Squares

Note that (a+b)(a+b) = (a+b)^{2} and (a-b)(a-b) = (a-b)^{2}.

(a+b)(a-b) = a^{2} - b^{2} is known as the difference of two squares. Often multiplying two numbers becomes easier when using algebraic expression. Let’s look at how to use them to help solve arithmetic problems.

Example

Find 57 \times 23

Write 57 as 50 + 7 and 23 as 20 + 3.

57 \times 23 = (50 + 7)(20 + 3)

= 50(20) + 50(3) + 7(20) + 7(3)

= 1000 + 150 + 140 + 21

= 1311

Or, we could have said 57 = 60 - 3 and 23 = 20 + 3, to obtain 57 \times 23 = (60 - 3)(20 + 3)

= 60(20) + 60(3) - 3(20) - 3(3)

= 1200 + 180 - 60 - 9

= 1311

Or, 57=60-3 and 23=30-7 to obtain:

57 \times 23 = (60-3)(30-7)

= 60(30) - 60(7) - 3(30) - 3(-7)

= 1800 - 420 - 90 + 21

= 1311


Example

Find { 63 }^{ 2 }

We solve this in two ways. First write 63 as 60 + 3. then use { (a + b) }^{ 2 } = { a }^{ 2 } + 2ab + { b }^{ 2 }. Also, we write 63 as 70 - 7 and use (a - b)^{ 2 } = { a }^{ 2 } - 2ab + { b }^{ 2 }.

We should arrive at the same conclusion:

{ 63 }^{ 2 } = { 60 + 3 }^{ 2 } = { 60 }^{ 2 } + 2(60) (3) + { 3 }^{ 2 }

= 3600 + 360 + 9

= 3969

Or

{ 63 }^{ 2 } = { 70 - 7 }^{ 2 } = { 70 }^{ 2 } - 2(70) (7) + { 7 }^{ 2 }

= 4900 - 980 + 49

= 3969, the same as the other answer above.


Example

Find 37 \times 43

37 \times 43 = (40 - 3) (40 + 3)

= { 40 }^{ 2 } - { 3 }^{ 2 }

= 1600 - 9

= 1591


Example

Find the value of the following

i) 202^{ 2 }

ii) 198^{ 2 }

iii) 202^{2} - 198^{2}

iv) 202 \times 198

i) 202^{ 2 } = (200 + 2)^{2}

= 200^{2} + 2(200)(2) + 2^{2}

= 40000 + 800 + 4

= 40804

ii) { 198 }^{ 2 } = { (200 - 2) }^{ 2 }

= 200^{2} - 2(200)(2) + 2^{2}

= 40000 - 800 + 4

= 39204

iii) 202^{2} - 198^{2} = 40804 - 39204

=1600

or, 202^{2} - 197^{2} = (202+198)(202-198)

=(400)(4)

=1600

iv) 202 \times 198 = (200 + 2) (200 - 2)

= { 200 }^{ 2 } - { 2 }^{ 2 }

= 40000 - 4

= 39996