### GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

# The Difference of Two Squares

The formula is known as the difference of two squares. It is often helpful for simplifying arithmetic problems by using algebraic expressions.

In the formula, and 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 and and the other with the difference between and . The right side of the equation represents the difference between the squares of and .

When multiplying the algebraic expression using the distributive property:   The middle terms, and , cancel each other out, resulting in the simplified form .

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: Notice that both expressions are in the form . In this case, and . Now, we can apply the difference of two squares formula: Calculate the squares: And subtract: Thus, the product of 97 and 103 is 9991.

Let’s look at some more examples.

## Examples

Example 1: Multiplying two numbers

Find Write 57 as and 23 as .    Alternatively, we could write and to get:    Example 2: Squaring a number

Find We can solve this in two ways. First, write 63 as , then use the formula . Alternatively, you could write 63 as and use the formula . Both methods should give the same result:   Or   , which is the same as the previous answer.

Example 3: Difference of squares in multiplication

Find     Example 4: Applying the difference of squares to various calculations

Find the value of the following:

i) ii) iii) iv) i)    ii)    iii)  Alternatively,   iv)    