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Perfect Squares

With the perfect squares formula, we learn how to write all the terms in the expansion of any binomial raised to the power of \(2\).
That's any expression looking like: \[\begin{pmatrix} a + b \end{pmatrix}^2 \quad \text{or} \quad \begin{pmatrix} a - b \end{pmatrix}^2\] We start by learning the formula and then we'll see some worked examples as well as how this formula can be used.

The Formula

The perfect square formula for expanding binomials raised to the power \(2\) are: \[\begin{pmatrix}a+b\end{pmatrix}^2 = a^2+2ab+b^2\]

and
\[\begin{pmatrix}a-b \end{pmatrix}^2=a^2-2ab+b^2\]

In the following tutorial, we review these formula as well as see how they can be used to write all the terms in the expansion of a binomial raised to the power of \(2\).

Tutorial

Example

Write all the terms in the expansions of each of the following:

  1. \(\begin{pmatrix} x + 4 \end{pmatrix}^2\)

  2. \(\begin{pmatrix} y-5 \end{pmatrix}^2\)

  3. \(\begin{pmatrix} 3y+2 \end{pmatrix}^2\)

  4. \(\begin{pmatrix} 6-3x \end{pmatrix}^2\)

  5. \(\begin{pmatrix} 3m+n \end{pmatrix}^2\)

  6. \(\begin{pmatrix}5p-3q \end{pmatrix}^2\)

  7. \(\begin{pmatrix}x^2-3 \end{pmatrix}\)

  8. \(\begin{pmatrix} 7x+y^3\end{pmatrix}^2\)

Solution Without Working

  1. \(x^2 - 8x + 16\)

  2. \(y^2 - 10y + 25\)

  3. \(9y^2 +12y + 4\)

  4. \(36 - 36x + 9y^2\)

  5. \(9m^2 + 6mn + n^2\)

  6. \(25p^2 - 30pq +9q^2\)

  7. \(x^4 - 6x^2 + 9\)

  8. \(49x^2 + 14xy^3 + y^6\)

Solution With Working

  1. The terms in the expansion of \(\begin{pmatrix} x+4 \end{pmatrix}^2\) are: \[\begin{aligned}\begin{pmatrix} x + 4 \end{pmatrix}^2 &= x^2-2.x.4+4^2 \\ & = x^2 - 8x + 16 \end{aligned}\]

  2. The terms in the expansion of \(\begin{pmatrix} y-5 \end{pmatrix}^2\) are: \[ \begin{aligned} \begin{pmatrix} y - 5 \end{pmatrix}^2 &= y^2 - 2.y.5 + 5^2 \\ & = y^2 - 10y + 25 \end{aligned}\]

  3. The terms in the expansion of \(\begin{pmatrix} 3y +2 \end{pmatrix}^2\) are: \[ \begin{aligned} \begin{pmatrix} 3y+2 \end{pmatrix}^2 &= \begin{pmatrix} 3y \end{pmatrix}^2 + 2.3y.2 + 2^2 \\ & = 3^2y^2+12y+4 \\ &= 9y^2+12y + 4 \end{aligned}\]

  4. The terms in the expansion of \(\begin{pmatrix} 6 - 3x \end{pmatrix}^2\) are: \[\begin{aligned} \begin{pmatrix} 6 - 3x \end{pmatrix}^2 &= 6^2 - 2.6.3x + \begin{pmatrix} 3x \end{pmatrix}^2 \\ &=36 - 36x + 3^2x^2 \\ & = 36 - 36x + 9x^2 \end{aligned}\]

  5. The terms in the expansion of \(\begin{pmatrix} 6 - 3x \end{pmatrix}^2\) are: \[\begin{aligned} \begin{pmatrix} 3m+n \end{pmatrix}^2 &= \begin{pmatrix} 3m \end{pmatrix}^2 + 2.3m.n + n^2 \\ & = 3^2m^2 + 6mn + n^2 \\ & = 9m^2 + 6mn + n^2 \end{aligned}\]

  6. The terms in the expansion of \(\begin{pmatrix} 5p-3q \end{pmatrix}^2\) are: \[\begin{aligned} \begin{pmatrix} 5p-3q \end{pmatrix}^2 &= = \begin{pmatrix} 5p \end{pmatrix}^2 - 2.5p.3q + \begin{pmatrix} 3q \end{pmatrix}^2 \\ & = 5^2p^2 -30pq +3^2q^2 \\ & = 25p^2 - 30pq +9q^2 \end{aligned}\]

  7. The terms in the expansion of \(\begin{pmatrix} x^2 - 3 \end{pmatrix}^2\) are: \[\begin{aligned} \begin{pmatrix} x^2 - 3 \end{pmatrix}^2 & = \begin{pmatrix} x^2 \end{pmatrix}^2 - 2.x^2.3 + 3^2 \\ & = x^{2\times 2} - 6x^2 + 9 \\ & = x^4 - 6x^2 + 9 \end{aligned}\]

  8. The terms in the expansion of \(\begin{pmatrix} 7x + y^3 \end{pmatrix}^2\) are: \[\begin{aligned} \begin{pmatrix} 7x + y^3 \end{pmatrix}^2 &= \begin{pmatrix} 7x \end{pmatrix}^2 + 2.7x.y^3 + \begin{pmatrix} y^3 \end{pmatrix}^2 \\ &= 7^2x^2 + 14xy^3+y^{3\times 2} \\ & = 49x^2 + 14xy^3 + y^6 \end{aligned}\]

Applications

Perfect squares have several applications.
We can also use the perfect square formula to make "quick calculations" like \(49^2\) and \(82^2\) or \(112^2\).
The "trick" is to write these numbers as perfect squares.

For instance, to calculate \(112^2\), we could write: \[\begin{aligned} 112^2 &= \begin{pmatrix}100 + 12 \end{pmatrix}^2 \\ & = 100^2+2\times 100\times 12 + 12^2 \\ & = 10000 + 2400 + 144 \\ & = 12400 + 144 \\ 122^2 & = 12544 \end{aligned}\] In the tutorial, below, we review how this is done with some examples, watch it now before working through the exercise further down.

In the following tutorial we see some worked examples, illustrating how to use perfect squares to do quick calculations without a calculator.

Tutorial

Example

Using the method we just saw and without using a calculator, calculate each of the following:

  1. \(101^2\)

  2. \(27^2\)

  3. \(23^2\)

  4. \(99^2\)

  5. \(104^2\)

  6. \(48^2\)

Solution Without Working

  1. \(101^2 = 10201\)

  2. \(27^2 = 711\)

  3. \(23^2 = 529\)

  4. \(99^2 = 9801\)

  5. \(104^2 = 10816\)

  6. \(48^2 = 2304\)

Solution With Working

  1. We calculate \(101^2\) as follows: \[\begin{aligned} 101^2 & = \begin{pmatrix} 100+1\end{pmatrix}^2 \\ & = 100^2 + 2 \times 100 \times 1 + 1^2 \\ & = 10000 + 200 + 1\\ 101^2 &= 10201 \end{aligned}\]

  2. We calculate \(27^2\) as follows: \[\begin{aligned} 27^2 & = \begin{pmatrix} 30 - 3 \end{pmatrix}^2 \\ & = 30^2 - 2 \times 30 \times 3 - 3^2 \\ & = 900 - 180 - 9 \\ & = 720 - 9 \\ 27^2 & = 711 \end{aligned}\]

  3. We calculate \(23^2\) as follows: \[\begin{aligned} 23^2 & = \begin{pmatrix} 20 + 3 \end{pmatrix}^2 \\ & = 20^2 + 2 \times 20 \times 3 + 3^2 \\ & = 400 + 120 + 9 \\ 23^2 & = 529 \end{aligned}\]

  4. We calculate \(99^2\) as follows: \[\begin{aligned} 99^2 & = \begin{pmatrix} 100 - 1 \end{pmatrix} \\ & = 100^2 - 2\times 100 \times 1 + 1^2 \\ & = 10000 - 200 + 1 \\ 99^2 & = 9800 + 1 \end{aligned}\]

  5. We calculate \(104^2\) as follows: \[\begin{aligned} 104^2 & = \begin{pmatrix} 100 + 4 \end{pmatrix}^2 \\ & = 100^2 + 2\times 100 \times 4 + 4^2 \\ & = 10000 + 800 + 16 \\ 104^2 & = 10816 \end{aligned}\]

  6. We calculate \(48^2\) as follows: \[\begin{aligned} 48^2 &= \begin{pmatrix} 50 - 2 \end{pmatrix}^2 \\ & = 50^2 - 2\times 50 \times 2 + 2^2 \\ & = 2500 - 200 + 4 \\ & = 2300 + 4 \\ 48^2 & = 2304 \end{aligned}\]