Q. 205 - Intermediate Algebra

Question

In the following exercises, multiply:

$$\begin{gathered} \left(a\right) {\left(3+\sqrt{5}\right)}^2 \\ \left(b\right) {\left(2-5\sqrt{3}\right)}^2 \end{gathered}$$

Step-by-Step Solution

Answer

Part a. By multiplying we get,$14+6\sqrt{5}.$

Part b. By multiplying we get, $79-20\sqrt{3}.$

Part a Step 1. Multiply the given expression

The given expression is ${\left(3+\sqrt{5}\right)}^2$

Multiply, using the Product of Binomial Squares Pattern 
$$\begin{gathered} {(3+\sqrt{5})}^2 \\ =3^2+2·3·\sqrt{5}+{\left(\sqrt{5}\right)}^2 \end{gathered}$$

Part a Step 2. Simplify

By simplifying we get,

$$\begin{gathered} =3^2+2·3·\sqrt{5}+{\left(\sqrt{5}\right)}^2 \\ =9+6\sqrt{5}+5 \\ =14+6\sqrt{5} \left[combine the like terms\right] \end{gathered}$$

Part b Step 1. Multiply the given expression

The given expression is ${\left(2-5\sqrt{3}\right)}^2.$

Multiply, using the Product of Binomial Squares Pattern

$$\begin{gathered} {(2-5\sqrt{3})}^2 \\ =2^2-2·2·5\sqrt{3}+{\left(5\sqrt{3}\right)}^2 \end{gathered}$$

Part b Step 2. Simplify

By simplifying we get,

$$\begin{gathered} =2^2-2·2·5\sqrt{3}+{\left(5\sqrt{3}\right)}^2 \\ =4-20\sqrt{3}+75 \\ =79-20\sqrt{3} \left[combine like terms\right] \end{gathered}$$