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Year 10 Interactive Maths - Second Edition


Perfect Squares

In algebra, expressions such as a squared and b squared are called perfect squares. (a + b) squared is also a perfect square which can be expanded to yield:

(a + b) squared equals a squared plus 2ab plus b squared.

Thus, a squared + 2ab + b squared is a perfect square since it can be written as the square of (a + b). That is:

(a + b) squared = a squared + 2ab + b squared.

is a perfect square.


Geometrical Illustration

Consider a square of side (a + b) units as shown in the following diagram:

The area is divided into 4 smaller areas of size a squared, b squared and two rectangles of size ab squared units.

Area of square ABCD = (a + b) squared

Also, area of square ABCD = sum of four rectangles = a squared + 2ab + b squared

From (1) and (2), we get:

(a + b) squared = a squared + 2ab + b squared

Likewise, (a - b) squared is a perfect square which can be expanded to yield a squared - 2ab + b squared.

Thus a squared - 2ab + b squared is a perfect square since it can be written as the square of a - b. That is:

(a - b) squared = a squared - 2ab + b squared

is a perfect square.


Applications

The formulas obtained above enable us to expand a perfect square quicker than by using the Distributive Law.

Use the perfect square formula to expand (x + 5) squared. The expansion equals x squared + 10x + 25.


Example 9

Expand these perfect squares using the perfect square formulas.

Solution:

The solution to Example 8 uses the perfect square formulas.


Key Terms

perfect square

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