Question 407874


{{{6a^2-9ab-15b^2}}} Start with the given expression



{{{3(2a^2-3ab-5b^2)}}} Factor out the GCF {{{3}}}



Now let's focus on the inner expression {{{2a^2-3ab-5b^2}}}





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Looking at {{{2a^2-3ab-5b^2}}} we can see that the first term is {{{2a^2}}} and the last term is {{{-5b^2}}} where the coefficients are 2 and -5 respectively.


Now multiply the first coefficient 2 and the last coefficient -5 to get -10. Now what two numbers multiply to -10 and add to the  middle coefficient -3? Let's list all of the factors of -10:




Factors of -10:

1,2,5,10


-1,-2,-5,-10 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -10

(1)*(-10)

(2)*(-5)

(-1)*(10)

(-2)*(5)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to -3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -3


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-10</td><td>1+(-10)=-9</td></tr><tr><td align="center">2</td><td align="center">-5</td><td>2+(-5)=-3</td></tr><tr><td align="center">-1</td><td align="center">10</td><td>-1+10=9</td></tr><tr><td align="center">-2</td><td align="center">5</td><td>-2+5=3</td></tr></table>



From this list we can see that 2 and -5 add up to -3 and multiply to -10



Now looking at the expression {{{2a^2-3ab-5b^2}}}, replace {{{-3ab}}} with {{{2ab+-5ab}}} (notice {{{2ab+-5ab}}} adds up to {{{-3ab}}}. So it is equivalent to {{{-3ab}}})


{{{2a^2+highlight(2ab+-5ab)+-5b^2}}}



Now let's factor {{{2a^2+2ab-5ab-5b^2}}} by grouping:



{{{(2a^2+2ab)+(-5ab-5b^2)}}} Group like terms



{{{2a(a+b)-5b(a+b)}}} Factor out the GCF of {{{2a}}} out of the first group. Factor out the GCF of {{{-5b}}} out of the second group



{{{(2a-5b)(a+b)}}} Since we have a common term of {{{a+b}}}, we can combine like terms


So {{{2a^2+2ab-5ab-5b^2}}} factors to {{{(2a-5b)(a+b)}}}



So this also means that {{{2a^2-3ab-5b^2}}} factors to {{{(2a-5b)(a+b)}}} (since {{{2a^2-3ab-5b^2}}} is equivalent to {{{2a^2+2ab-5ab-5b^2}}})




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So our expression goes from {{{3(2a^2-3ab-5b^2)}}} and factors further to {{{3(2a-5b)(a+b)}}}



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Answer:


So {{{6a^2-9ab-15b^2}}} factors to {{{3(2a-5b)(a+b)}}}

    

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