Chapter 4.5, Problem 32WE

To determine

To simplify the equation

  $\frac{1}{2}\left(12-4s^2\right)-2s\left(1-s\right)=8$

Answer to Problem 32WE

  $s=-1$

Explanation of Solution

Given:

  $\frac{1}{2}\left(12-4s^2\right)-2s\left(1-s\right)=8$

Concept Used:

• To simplify an expression first use distributive property to remove any grouping symbol, and then add/subtract like terms.
• Distributive property: $a\left(b\pm c\right)=ab\pm ac$
• If an expression contains more than one operations between its terms and brackets, first solve the terms inside brackets and then apply division followed by multiplication, addition and subtraction.

Calculation:

To simplify the equation

  $\frac{1}{2}\left(12-4s^2\right)-2s\left(1-s\right)=8$

First use distributive property to remove any grouping symbol as,

  $\begin{array}{l}\frac{1}{2}\left(12-4s^2\right)-2s\left(1-s\right)=8\\ \Rightarrow \frac{1}{2}\left(12\right)-\frac{1}{2}\left(4s^2\right)-2s\left(1\right)-2s\left(-s\right)=8\\ \Rightarrow 6-2s^2-2s+2s^2=8\\ \Rightarrow 6-2s^2+2s^2-2s=8\\ \Rightarrow 6-2s=8\end{array}$

Now, subtract both sides of the equation by $6$ as,

  $\begin{array}{l}6-6-2s=8-6\\ \Rightarrow -2s=2\\ \Rightarrow -1\left(-2s\right)=2\left(-1\right)\text{ by multiplying both sides by }-1\\ \Rightarrow 2s=-2\\ \Rightarrow \frac{2s}{2}=\frac{-2}{2}\text{ by dividing both sides by }2\\ \Rightarrow s=-1\end{array}$

Thus, the simplified equation becomes

  $s=-1$