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Square of a Binomial

The square of a binomial is always a trinomial. It will be helpful to memorize these patterns for writing squares of binomials as trinomials.

${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$

${(a - b)}^{2} = a^{2} - 2 a b + b^{2}$

Examples:

Square each binomial.

$a) {(x + 4)}^{2}$

$\begin{array}{l} {(x + 4)}^{2} = x^{2} + 2 (x \cdot 4) + 4^{2} \\ = x^{2} + 8 x + 16 \end{array}$

$b) {(2 y - 3)}^{2}$

$\begin{array}{l} {(2 y - 3)}^{2} = {(2 y)}^{2} - 2 (2 y \cdot 3) + 3^{2} \\ = {(2 y)}^{2} - 2 (6 y) + 3^{2} \\ = 4 y^{2} - 12 y + 9 \end{array}$

$c) {(3 p - 2 q^{2})}^{2}$

$\begin{array}{l} {(3 p - 2 q^{2})}^{2} = {(3 p)}^{2} - 2 (3 p \cdot 2 q^{2}) + {(2 q^{2})}^{2} \\ = 9 p^{2} - 2 (6 p q^{2}) + 4 q^{4} \\ = 9 p^{2} - 12 p q^{2} + 4 q^{4} \end{array}$

If the coefficients of a trinomial $a x^{2} + b x + c$ satisfy the equation

$c = {(\frac{b}{2 \sqrt{a}})}^{2}$ ,

then the trinomial is the perfect square of the binomial

$\sqrt{a} x + \frac{b}{2}$ .

Example 1:

Factor, if possible.

$x^{2} - 14 x + 49$

Here, $a = 1, b = - 14, and c = 49$ . We have:

${(\frac{b}{2 \sqrt{a}})}^{2} = {(\frac{- 14}{2 \sqrt{1}})}^{2} = {(- 7)}^{2} = 49 = c$

So, the trinomial is a perfect square:

$x^{2} - 14 x + 49 = {(x - 7)}^{2}$

You can verify this using FOIL .

Example 2:

Factor, if possible.

$9 w^{4} + 12 w^{2} + 4$

Here, $a = 9, b = 12, and c = 4$ . (We can treat $w^{2}$ as $x$ , and not worry about the fourth power.)

${(\frac{b}{2 \sqrt{a}})}^{2} = {(\frac{12}{2 \sqrt{9}})}^{2} = {(\frac{12}{6})}^{2} = 4 = c$

So, the trinomial is a perfect square:

$9 w^{4} + 12 w^{2} + 4 = {(3 w^{2} + 2)}^{2}$

This can also be verified using FOIL.