Chapter 8.4, Problem 61HP

To determine

To determine the square the sum of two quantities, square the difference of two quantities, and how to find the product of a sum of two quantities and a difference of two quantities.

Answer to Problem 61HP

  ${(a+b)}^2=a^2+2ab+b^2$

  ${(a-b)}^2=a^2-2ab+b^2$

  $(a+b)(a-b)=a^2-b^2$

Explanation of Solution

Given:

The square the sum of two quantities; square the difference of two quantities; product of a sum of two quantities and a difference of two quantities.

Formula used:

  ${(a+b)}^2=a^2+2ab+b^2$

  ${(a-b)}^2=a^2-2ab+b^2$

  $(a+b)(a-b)=a^2-b^2$

Calculation:

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

  $\Rightarrow (a+b)(a+b)$

This can also be represented as,

  $\Rightarrow (a+b)(a+b)$

  $\Rightarrow {(a+b)}^2$

Now,

  $\Rightarrow {(a+b)}^2$

  $\Rightarrow a^2+2ab+b^2$  [ $\because$ ${(a+b)}^2=a^2+2ab+b^2$ ]

The square difference of two quantities can be represented as,

  $\Rightarrow (a-b)(a-b)$

This can also be represented as,

  $\Rightarrow (a-b)(a-b)$

  $\Rightarrow {(a-b)}^2$

Now,

  $\Rightarrow {(a-b)}^2$

  $\Rightarrow a^2-2ab+b^2$  [ $\because$ ${(a-b)}^2=a^2-2ab+b^2$ ]

The product of a sum of two quantities and a difference of two quantities can be represented as,

  $\Rightarrow (a+b)(a-b)$

This can also be represented as,

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

Now, removing $-ab+ab$

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

Conclusion:

Therefore, the square the sum of two quantities is ${(a+b)}^2=a^2+2ab+b^2$ ; the square difference of two quantities is ${(a-b)}^2=a^2-2ab+b^2$ ; the product of a sum of two quantities and a difference of two quantities is $(a+b)(a-b)=a^2-b^2$ .