EDGE Math 8

Squares of Binomials

To find the square of a sum of two terms, ${(a + b)}^{2},$ write it as $(a + b) (a + b) .$ Then apply the FOIL method and combine like terms. You will get the following:

\begin{array}{l} {(a + b)}^{2} = (a + b) (a + b) \\ = a^{2} + a b + a b + b^{2} \\ = a^{2} + 2 a b + b^{2} \end{array}

The same steps may be done to find the square of a difference of two terms, ${(a - b)}^{2} .$ Thus, you will have the following equations:

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

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

Example

Multiply each of the following:

1. ${(3 x + y)}^{2}$

2. ${(\frac{1}{2} p + 3 q)}^{2}$

3. ${(5 x y - 1)}^{2}$

Solution

{(3 x + y)}^{2}

= {(3 x)}^{2} + 2 (3 x) (y) + y^{2}

= 9 x^{2} + 6 x y + y^{2}

{(\frac{1}{2} p + 3 q)}^{2}

= {(\frac{1}{2} p)}^{2} + 2 (\frac{1}{2} p) (3 q) + {(3 q)}^{2}

= \frac{p^{2}}{4} + 3 p q + 9 q^{2}

{(5 x y - 1)}^{2}

= {(5 x y)}^{2} - 2 (5 x y) (1) + {(- 1)}^{2}

= 25 x^{2} y^{2} - 10 x y + 1