Question

Using the identities, simplify the following :

${4(\frac{2}{3}p-\frac{3}{4}q)}^2$

Answer 1

Step 1: Identification of identity

First we need to identify the identity we can use,

Here we need to find the square, so we can use ${(a-b)}^2=a^2-2ab+b^2$

Now if we compare the question with the identity we get $a=\frac{2}{3}p,b=\frac{3}{4}q$

Step 2: Applying the formula

$$\begin{gathered} {4(\frac{2}{3}p-\frac{3}{4}q)}^2={4\left[\right(\frac{2}{3}p)}^2-2\left(\frac{2}{3}p\right)\left(\frac{3}{4}q\right)+{\left(\frac{3}{4}q\right)}^2] \\ =4[\frac{4}{9}{p}^2-pq+\frac{9}{16}{q}^2] \\ =\frac{16}{9}{p}^2-4pq+\frac{9}{4}{q}^2 \end{gathered}$$

Hence, the square of ${4(\frac{2}{3}p-\frac{3}{4}q)}^2$ is $\frac{16}{9}{p}^2-4pq+\frac{9}{4}{q}^2$.