A3.2: Perfect squares

The idea of a perfect square permeates mathematics and especially algebra. Properties of perfect squares will help you in factorising and expanding algebraic expressions. An understanding of perfect squares is essential for success in mathematics.

A perfect square is something like \(5^{2}, x^{2}\) or \(\left(x+1\right)^{2}\). In this module we deal with perfect squares of the form: \(\left(x\pm y\right)^{2} =x^{2}\pm2xy+y^{2}.\)These forms are very common in mathematics and help us simplify expressions. They are important to know if you are an engineering or science student.

Is this quadratic a perfect square? Some quadratics (but not all) are perfect squares. This means they can be factorised neatly into the form of (a+b) (a-b). Find out if your quadratic can be factorised in this simple way.

Examples of perfect squares are \(5^{2},\,x^{2},\,a^{2}b^{2},\,\left(xy\right)^{2}\) and \(\left(a\pm b\right)^{2}.\) For your development of algebraic skills, the most interesting of these are \(\left(a+b\right)^{2}\) and \(\left(a-b\right)^{2}.\)

Rules for expanding perfect squares

Consider \(\left(a+b\right)^{2}.\) We have \[\begin{align*} \left(a+b\right)^{2} & =\left(a+b\right)\left(a+b\right)\text{ by definition}\\ & =a^{2}+ab+ba+b^{2}\\ & =a^{2}+2ab+b^{2}. & \left(1\right) \end{align*}\] Similarly, \[\begin{align*} \left(a-b\right)^{2} & =\left(a-b\right)\left(a-b\right)\text{ by definition}\\ & =a^{2}-ab-ba+b^{2}\\ & =a^{2}-2ab+b^{2}. & \left(2\right) \end{align*}\] You must remember the equations \(\left(1\right)\) and \(\left(2\right)\). That is

\[\begin{align*} \left(a+b\right)^{2} & =a^{2}+2ab+b^{2} \end{align*}\]

and

\[\begin{align*} \left(a-b\right)^{2} & =a^{2}-2ab+b^{2}. \end{align*}\]

Note that

• the first and last terms must be positive and must be perfect squares.
• the middle term must be twice the product of the first and last terms and may be positive or negative.

Using perfect squares for factorising

The rules for expanding perfect squares may be used in reverse to factorise algebraic expressions.

Example 1

Factorise \(x^{2}+14x+49.\)

Solution:

Note that \(49=7^{2}\) and \(14=2\times7\). Applying equation \(\left(1\right)\)above, \[\begin{align*} x^{2}+14x+49 & =\left(x+7\right)^{2}. \end{align*}\] And \(x^{2}+14x+49\) is a perfect square.

Example 2

Is \(y^{2}-20y+25\) a perfect square?

Solution

Note that \(25=5^{2}\) but \(-20\neq2\times5\) hence \(y^{2}-20y+25\) is not a perfect square.

Example 3

Is \(4a^{2}-12a-9\) a perfect square?

Solution

The last term is negative and so \(4a^{2}-12a-9\) is not a perfect square.

Example 4

Is \(100x^{2}-180x+81\) a perfect square?

Solution

\[\begin{align*} 100x^{2} & =\left(10x\right)^{2}\\ 81 & =9^{2} \end{align*}\] and \(180=2\times10\times9.\) Hence

\[\begin{align*} 100x^{2}-180x+81 & =\left(10x-9\right)^{2} \end{align*}\] and is a perfect square.

Example 5

Factorise \(50x^{2}+80x+32\) ?

Solution

At first sight, the expression is not a perfect square because neither \(50x^{2}\) nor \(32\) is a perfect square. However, if we divide the expression by \(2\) we have \[\begin{align*} 50x^{2}+80x+32 & =2\left(25x^{2}+40x+16\right) & \left(3\right) \end{align*}\] The expression \(25x^{2}+40x+16\) is a perfect square because \[\begin{align*} 25x^{2}+40x+16 & =\left(5x+4\right)^{2} \end{align*}\] and so from equation \(\left(3\right)\) \[\begin{align*} 50x^{2}+80x+32 & =2\left(5x+4\right)^{2}. \end{align*}\] The answer is that \(50x^{2}+80x+32\) is not a perfect square. However \(50x^{2}+80x+32=2\left(5x+4\right)^{2}\) which is a useful factorisation.

Exercise

Check each of the following expressions. If it is a perfect square, state the perfect square.

\(1.\;a^{2}+2a+1\)

\(2.\ x^{2}-4x+4\)

\(3.\ 25x^{2}-10x+1\)

\(4.\ 4y^{2}-6y+9\)

\(\text{5.$\;81x^{2}+108x+36$ }\)

\(\text{6.$\;9a^{2}-24a-16$ }\)

\(\text{7.$\;16x^{2}-40xy+25y^{2}$ }\)

\(\text{8.$\;121z^{2}+88z+64$ }\)

\(\text{9.$\;2x^{2}+8x+8\,.$ }\)