The Difference of Two Squares

The formula (a+b)(a-b) = a^{2} - b^{2} is known as the difference of two squares. It is often helpful for simplifying arithmetic problems by using algebraic expressions.

In the formula, a and b represent any real numbers, variables, or expressions. The left side of the equation is an algebraic expression with two parts, one with the sum of a and b and the other with the difference between a and b. The right side of the equation represents the difference between the squares of a and b.

When multiplying the algebraic expression using the distributive property:

(a + b)(a - b) = a(a - b) + b(a - b)

= a^2 - ab + ab - b^2

= a^2 - b^2

The middle terms, -ab and +ab, cancel each other out, resulting in the simplified form a^2 - b^2.

Let’s look at an example to see how this can be applied to a calculation

Example: Multiply 97 and 103

Using the difference of two squares formula, we can rewrite this multiplication as:

97 \times 103 = (100 - 3)(100 + 3)

Notice that both expressions are in the form (a + b)(a - b). In this case, a = 100 and b = 3. Now, we can apply the difference of two squares formula:

(100 - 3)(100 + 3) = 100^2 - 3^2

Calculate the squares:

= 10000 - 9

And subtract:

= 9991

Thus, the product of 97 and 103 is 9991.

Let’s look at some more examples.

Examples

Example 1: Multiplying two numbers

Find 57 \times 23

Write 57 as 50 + 750 + 7 and 23 as 20 + 320 + 3.

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

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

= 1000 + 150 + 140 + 21= 1000 + 150 + 140 + 21

= 1311= 1311

Alternatively, we could write 57 = 60 - 357 = 60 - 3 and 23 = 20 + 323 = 20 + 3 to get:

57 \times 23 = (60 - 3)(20 + 3)57 \times 23 = (60 - 3)(20 + 3)

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

= 1200 + 180 - 60 - 9= 1200 + 180 - 60 - 9

= 1311= 1311

Example 2: Squaring a number

Find {63}^{2}

We can solve this in two ways. First, write 63 as 60 + 360 + 3, then use the formula {(a + b)}^{2} = {a}^{2} + 2ab + {b}^{2}{(a + b)}^{2} = {a}^{2} + 2ab + {b}^{2}. Alternatively, you could write 63 as 70 - 770 - 7 and use the formula {(a - b)}^{2} = {a}^{2} - 2ab + {b}^{2}{(a - b)}^{2} = {a}^{2} - 2ab + {b}^{2}. Both methods should give the same result:

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

= 3600 + 360 + 9= 3600 + 360 + 9

= 3969= 3969

Or

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

= 4900 - 980 + 49= 4900 - 980 + 49

= 3969= 3969, which is the same as the previous answer.

Example 3: Difference of squares in multiplication

Find 37 \times 43

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

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

= 1600 - 9= 1600 - 9

= 1591= 1591

Example 4: Applying the difference of squares to various calculations

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

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

= 40000 + 800 + 4= 40000 + 800 + 4

= 40804= 40804

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

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

= 40000 - 800 + 4= 40000 - 800 + 4

= 39204= 39204

iii) 202^{2} - 198^{2} = 40804 - 39204202^{2} - 198^{2} = 40804 - 39204

= 1600= 1600

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

= (400)(4)= (400)(4)

= 1600= 1600

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

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

= 40000 - 4= 40000 - 4

= 39996= 39996

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