Question 195512
# 1




Looking at {{{x^2-9xy+14y^2}}} we can see that the first term is {{{x^2}}} and the last term is {{{14y^2}}} where the coefficients are 1 and 14 respectively.


Now multiply the first coefficient 1 and the last coefficient 14 to get 14. Now what two numbers multiply to 14 and add to the  middle coefficient -9? Let's list all of the factors of 14:




Factors of 14:

1,2,7,14


-1,-2,-7,-14 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 14

1*14

2*7

(-1)*(-14)

(-2)*(-7)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">14</td><td>1+14=15</td></tr><tr><td align="center">2</td><td align="center">7</td><td>2+7=9</td></tr><tr><td align="center">-1</td><td align="center">-14</td><td>-1+(-14)=-15</td></tr><tr><td align="center">-2</td><td align="center">-7</td><td>-2+(-7)=-9</td></tr></table>



From this list we can see that -2 and -7 add up to -9 and multiply to 14



Now looking at the expression {{{x^2-9xy+14y^2}}}, replace {{{-9xy}}} with {{{-2xy-7xy}}} (notice {{{-2xy-7xy}}} combines to {{{-9xy}}}. So it is equivalent to {{{-9xy}}})


{{{x^2+highlight(-2xy-7xy)+14y^2}}}



Now let's factor {{{x^2-2xy-7xy+14y^2}}} by grouping:



{{{(x^2-2xy)+(-7xy+14y^2)}}} Group like terms



{{{x(x-2y)-7y(x-2y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-7y}}} out of the second group



{{{(x-7y)(x-2y)}}} Since we have a common term of {{{x-2y}}}, we can combine like terms


So {{{x^2-2xy-7xy+14y^2}}} factors to {{{(x-7y)(x-2y)}}}



So this also means that {{{x^2-9xy+14y^2}}} factors to {{{(x-7y)(x-2y)}}} (since {{{x^2-9xy+14y^2}}} is equivalent to {{{x^2-2xy-7xy+14y^2}}})




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     Answer:

So {{{x^2-9xy+14y^2}}} factors to {{{(x-7y)(x-2y)}}}



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# 2




Looking at {{{6a^2-5ab-4b^2}}} we can see that the first term is {{{6a^2}}} and the last term is {{{-4b^2}}} where the coefficients are 6 and -4 respectively.


Now multiply the first coefficient 6 and the last coefficient -4 to get -24. Now what two numbers multiply to -24 and add to the  middle coefficient -5? Let's list all of the factors of -24:




Factors of -24:

1,2,3,4,6,8,12,24


-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -24

(1)*(-24)

(2)*(-12)

(3)*(-8)

(4)*(-6)

(-1)*(24)

(-2)*(12)

(-3)*(8)

(-4)*(6)


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



Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-24</td><td>1+(-24)=-23</td></tr><tr><td align="center">2</td><td align="center">-12</td><td>2+(-12)=-10</td></tr><tr><td align="center">3</td><td align="center">-8</td><td>3+(-8)=-5</td></tr><tr><td align="center">4</td><td align="center">-6</td><td>4+(-6)=-2</td></tr><tr><td align="center">-1</td><td align="center">24</td><td>-1+24=23</td></tr><tr><td align="center">-2</td><td align="center">12</td><td>-2+12=10</td></tr><tr><td align="center">-3</td><td align="center">8</td><td>-3+8=5</td></tr><tr><td align="center">-4</td><td align="center">6</td><td>-4+6=2</td></tr></table>



From this list we can see that 3 and -8 add up to -5 and multiply to -24



Now looking at the expression {{{6a^2-5ab-4b^2}}}, replace {{{-5ab}}} with {{{3ab+-8ab}}} (notice {{{3ab+-8ab}}} adds up to {{{-5ab}}}. So it is equivalent to {{{-5ab}}})


{{{6a^2+highlight(3ab+-8ab)+-4b^2}}}



Now let's factor {{{6a^2+3ab-8ab-4b^2}}} by grouping:



{{{(6a^2+3ab)+(-8ab-4b^2)}}} Group like terms



{{{3a(2a+b)-4b(2a+b)}}} Factor out the GCF of {{{3a}}} out of the first group. Factor out the GCF of {{{-4b}}} out of the second group



{{{(3a-4b)(2a+b)}}} Since we have a common term of {{{2a+b}}}, we can combine like terms


So {{{6a^2+3ab-8ab-4b^2}}} factors to {{{(3a-4b)(2a+b)}}}



So this also means that {{{6a^2-5ab-4b^2}}} factors to {{{(3a-4b)(2a+b)}}} (since {{{6a^2-5ab-4b^2}}} is equivalent to {{{6a^2+3ab-8ab-4b^2}}})




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     Answer:

So {{{6a^2-5ab-4b^2}}} factors to {{{(3a-4b)(2a+b)}}}