Question

Find each of the following products:
(i) (x + 6)(x + 6)
(ii) (4x + 5y)(4x + 5y)
(iii) (7a + 9b)(7a + 9b)
(iv) $\left(\frac{2}{3}x+\frac{4}{5}y\right)\left(\frac{2}{3}x+\frac{4}{5}y\right)$
(v) (x² + 7)(x² + 7)
(vi) $\left(\frac{5}{6}{a}^2+2\right)\left(\frac{5}{6}{a}^2+2\right)$

Answer 1

(i) We have:

$$\begin{gathered} (x+6)(x+6) \\ =(x+6{)}^2 \\ =x^2+6^2+2\times x\times 6[\mathrm{using}(a+b{)}^2=a^2+b^2+2ab] \\ =x^2+36+12x \end{gathered}$$

(ii) We have:

$$\begin{gathered} (4x+5y)(4x+5y) \\ =(4x+5y{)}^2 \\ ={\left(4x\right)}^2+{\left(5y\right)}^2+2\times 4x\times 5y[\mathrm{using}(a+b{)}^2=a^2+b^2+2ab] \\ =16x^2+25y^2+40xy \end{gathered}$$

(iii) We have:
$$\begin{gathered} (7a+9b)(7a+9b) \\ =(7a+9b{)}^2 \\ ={\left(7a\right)}^2+{\left(9b\right)}^2+2\times 7a\times 9b[\mathrm{using}(a+b{)}^2=a^2+b^2+2ab] \\ =49a^2+81b^2+126ab \end{gathered}$$

(iv) We have:
$$\begin{gathered} \left(\frac{2}{3}x+\frac{4}{5}y\right)\left(\frac{2}{3}x+\frac{4}{5}y\right) \\ ={\left(\frac{2}{3}x+\frac{4}{5}y\right)}^2 \\ ={\left(\frac{2}{3}x\right)}^2+{\left(\frac{4}{5}y\right)}^2+2\times \frac{2}{3}x\times \frac{4}{5}y[\mathrm{using}(a+b{)}^2=a^2+b^2+2ab] \\ =\frac{4}{9}{x}^2+\frac{16}{25}{y}^2+\frac{16}{15}xy \end{gathered}$$


(v) We have:
$$\begin{gathered} (x^2+7)(x^2+7) \\ =(x^2+7{)}^2 \\ ={\left(x^2\right)}^2+7^2+2\times x^2\times 7[\mathrm{using}(a+b{)}^2=a^2+b^2+2ab] \\ =x^4+49+14x^2 \end{gathered}$$

(vi) We have:
$$\begin{gathered} \left(\frac{5}{6}{a}^2+2\right)\left(\frac{5}{6}{a}^2+2\right) \\ ={\left(\frac{5}{6}{a}^2+2\right)}^2 \\ ={\left(\frac{5}{6}{a}^2\right)}^2+{\left(2\right)}^2+2\times \frac{5}{6}{a}^2\times 2[\mathrm{using}(a+b{)}^2=a^2+b^2+2ab] \\ =\frac{25}{36}{a}^4+4+\frac{10}{3}{a}^2 \end{gathered}$$