<PRE><B>Question 299484</B>
Note: {{{9x^2-25}}} can be written as {{{9x^2+0x-25}}}



Looking at the expression {{{9x^2+0x-25}}}, we can see that the first coefficient is {{{9}}}, the second coefficient is {{{0}}}, and the last term is {{{-25}}}.



Now multiply the first coefficient {{{9}}} by the last term {{{-25}}} to get {{{(9)(-25)=-225}}}.



Now the question is: what two whole numbers multiply to {{{-225}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{0}}}?



To find these two numbers, we need to list &lt;font size=4&gt;&lt;b&gt;all&lt;/b&gt;&lt;/font&gt; of the factors of {{{-225}}} (the previous product).



Factors of {{{-225}}}:

1,3,5,9,15,25,45,75,225

-1,-3,-5,-9,-15,-25,-45,-75,-225



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-225}}}.

1*(-225) = -225
3*(-75) = -225
5*(-45) = -225
9*(-25) = -225
15*(-15) = -225
(-1)*(225) = -225
(-3)*(75) = -225
(-5)*(45) = -225
(-9)*(25) = -225
(-15)*(15) = -225


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{0}}}:



&lt;table border=&quot;1&quot;&gt;&lt;th&gt;First Number&lt;/th&gt;&lt;th&gt;Second Number&lt;/th&gt;&lt;th&gt;Sum&lt;/th&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-225&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-225)=-224&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-75&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+(-75)=-72&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-45&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5+(-45)=-40&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;9+(-25)=-16&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;15+(-15)=0&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;225&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+225=224&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;75&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+75=72&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;45&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5+45=40&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-9+25=16&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-15+15=0&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-15}}} and {{{15}}} add to {{{0}}} (the middle coefficient).



So the two numbers {{{-15}}} and {{{15}}} both multiply to {{{-225}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{0}}}



Now replace the middle term {{{0x}}} with {{{-15x+15x}}}. Remember, {{{-15}}} and {{{15}}} add to {{{0}}}. So this shows us that {{{-15x+15x=0x}}}.



{{{9x^2+highlight(-15x+15x)-25}}} Replace the second term {{{0x}}} with {{{-15x+15x}}}.



{{{(9x^2-15x)+(15x-25)}}} Group the terms into two pairs.



{{{3x(3x-5)+(15x-25)}}} Factor out the GCF {{{3x}}} from the first group.



{{{3x(3x-5)+5(3x-5)}}} Factor out {{{5}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(3x+5)(3x-5)}}} Combine like terms. Or factor out the common term {{{3x-5}}}



===============================================================



Answer:



So {{{9x^2+0x-25}}} factors to {{{(3x+5)(3x-5)}}}.



In other words, {{{9x^2+0x-25=(3x+5)(3x-5)}}}.



So {{{9x^2-25=(3x+5)(3x-5)}}}.



Note: you can check the answer by expanding {{{(3x+5)(3x-5)}}} to get {{{9x^2+0x-25}}} or by graphing the original expression and the answer (the two graphs should be identical). Also, {{{9x^2-25}}} is a difference of squares.
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