Section 2.3
The Differences of Two Squares
Since the word difference means "subtract", the phrase "difference of two squares" means that one perfect square will be subtracted from another. As we will see, the base of these perfect squares can be either numbers or expressions.
The difference of two squares (an expression of the form ) always factors. The factored form is $$A^2-B^2=(A+B)(A-B)$$
Examples:
a. $x^2-4=(x+2)(x-2)$ | The
base of the first term is $x$. The base of the second term is 2. |
b. $K^2-9=(K+9)(K-9)$ |
The
base of the first term is $K$. The base of the second term is 3. |
c. \begin{align} 4t^2-25&=(2t)^2-25\\ &=(2t+5)(2t-5) \end{align} |
The base of the first term is $2t$. The base of the second term is 5. |
d.
$(a+b)^2-r^2=\left[(a+b)+r\right]\left[(a+b)-r\right]$ |
The
base of the first term is $(a+b)$. The base of the second term is
$r$. |
e. \begin{align} x^2-4x+4-y^2&=\left[x^2-4x+4\right]-y^2\\ &=(x-2)^2-y^2\\ &=\left[(x-2)+y\right]\left[(x-2)-y\right] \end{align} |
Grouping $x^2-4x+4$ together we recognize that this is a perfect trinomial square. So, $x^2-4x+4=(x-2)^2$. Therefore the base of the first term is $(x-2)$. The base of the second term is $y$. |
Since factorization of expressions is similar to numbers, we have the following useful kind of multiplication problems.
\begin{align}
17\cdot23&=(20-3)\cdot(20+3)\\
&=20^2-3^3\\
&=400-9\\
&=391
\end{align}
\begin{align}
26\cdot 34&=(30-4)\cdot(30+4)\\
&=30^2-4^2\\
&=900-16\\
&=884
\end{align}
This can be done any time we have the product of two odd numbers or two even numbers.
It is important to remember that the sum of two squares rarely factors; there is no factorization rule for $A^2+B^2$ .
Also, you should always start any factoring problem by looking for a common factor.
Example 1: Factoring difference of squares: Factoring difference of squares
Example 2: Factoring to produce difference of squares:
Exercises:
Factor each of the following completely, if possible.
\begin{array}{l l l}
1. \;\;{{x}^{2}}-36\;\;& 2. \;\;{{x}^{2}}-81\;\;& 3. \;\;2{{x}^{2}}-50\;\;\\
4. \;\;{{x}^{2}}-9{{t}^{2}}\;\;& 5. \;\;32{{x}^{2}}-18\;\;& 6. \;\;9{{x}^{2}}-81\;\;\\
7. \;\;{{(a+3)}^{2}}-{{t}^{2}}\;\;& 8. \;\;4{{x}^{2}}{{y}^{2}}-1\;\;& 9. \;\;{{x}^{2}}+9\;\;\\
10. \;\;{{(2a-5)}^{2}}-4{{K}^{2}}\;\;& 11. \;\;4{{x}^{2}}+16{{y}^{2}}\;\;& 12. \;\;5{{x}^{3}}-125x\;\;\\
13. \;\;{{(x+4)}^{2}}-{{(y-4)}^{2}}\;\;& 14. \;\;3{{x}^{2}}-27\;\;& 15. \;\;5{{(x-y)}^{2}}-20{{K}^{2}}\;\;\\
16. \;\;3{{(x-y)}^{2}}+27{{K}^{2}}\;\;& 17. \;\;48{{K}^{2}}-12\;\;& 18. \;\;{{K}^{2}}-169\;\;\\
19. \;\;12{{t}^{4}}-3\;\;& 20. \;\;121{{t}^{2}}-{{t}^{4}}\;\;& 21. \;\;25{{a}^{2}}-1\;\;\\
22. \;\;{{x}^{4}}-1\;\;& 23. \;\;{{x}^{4}}-4\;\;& 24. \;\;{{x}^{4}}-16\;\;\\
25. \;\;1-{{(x-y)}^{2}}\;\;& 26. \;\;{{50}^{2}}-{{3}^{2}}\;\;& 27. \;\;{{500}^{2}}-{{30}^{2}}\;\;\\
28. \;\;{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{2}}-4{{x}^{2}}{{y}^{2}}\;\;& 29. \;\;{{\left( {{x}^{2}}+4{{y}^{2}} \right)}^{2}}-16{{x}^{2}}{{y}^{2}}\;\;& 30. \;\;{{\left( {{x}^{2}}+9{{y}^{2}} \right)}^{2}}-36{{x}^{2}}{{y}^{2}}\;\;
\end{array}