Chapter 8.4, Problem 64PFA

To determine

To determine the pattern use to find the square of the sum and the square of the difference differ.

Answer to Problem 64PFA

Differ by $-2ab$ and $+2ab$

Explanation of Solution

Given:

The square of the sum and the square of the difference.

Formula used:

  ${(a+b)}^2=a^2+2ab+b^2$

  ${(a-b)}^2=a^2-2ab+b^2$

Calculation:

The square of the sum of two quantities can be represented as,

  $\Rightarrow (a+b)(a+b)$

This can also be represented as,

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

Now,

  $\begin{array}{l}$ \Rightarrow {(a+b)}^2$\Rightarrow a^2+2ab+b^2\to \text{(i)}\end{array}$ [ $\because$ ${(a+b)}^2=a^2+2ab+b^2$ ]

The square difference of two quantities can be represented as,

  $\Rightarrow (a-b)(a-b)$

This can also be represented as,

  $\begin{array}{l}\Rightarrow (a-b)(a-b)\\ $ \Rightarrow {(a+b)}^2$\end{array}$

Now,

  $\begin{array}{l}$ \Rightarrow {(a+b)}^2$\Rightarrow a^2-2ab+b^2\to \text{(ii)}\end{array}$ [ $\because$ ${(a-b)}^2=a^2-2ab+b^2$ ]

Difference in their pattern can be determined by comparing equation $(i)$ and $(ii)$ ,

The only difference is in equation $(i)$ there is $+2ab$ but in equation $(ii)$ there is $-2ab$ .

Conclusion:

Therefore, the pattern use to find the square of the sum and the square of the difference differ by $-2ab$ and $+2ab$ .