Question 115752
#1

Looking at {{{4x^2-12x-40}}} we can see that the first term is {{{4x^2}}} and the last term is {{{-40}}} where the coefficients are 4 and -40 respectively.


Now multiply the first coefficient 4 and the last coefficient -40 to get -160. Now what two numbers multiply to -160 and add to the  middle coefficient -12? Let's list all of the factors of -160:




Factors of -160:

1,2,4,5,8,10,16,20,32,40,80,160


-1,-2,-4,-5,-8,-10,-16,-20,-32,-40,-80,-160 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -160

(1)*(-160)

(2)*(-80)

(4)*(-40)

(5)*(-32)

(8)*(-20)

(10)*(-16)

(-1)*(160)

(-2)*(80)

(-4)*(40)

(-5)*(32)

(-8)*(20)

(-10)*(16)


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



Now which of these pairs add to -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-160</td><td>1+(-160)=-159</td></tr><tr><td align="center">2</td><td align="center">-80</td><td>2+(-80)=-78</td></tr><tr><td align="center">4</td><td align="center">-40</td><td>4+(-40)=-36</td></tr><tr><td align="center">5</td><td align="center">-32</td><td>5+(-32)=-27</td></tr><tr><td align="center">8</td><td align="center">-20</td><td>8+(-20)=-12</td></tr><tr><td align="center">10</td><td align="center">-16</td><td>10+(-16)=-6</td></tr><tr><td align="center">-1</td><td align="center">160</td><td>-1+160=159</td></tr><tr><td align="center">-2</td><td align="center">80</td><td>-2+80=78</td></tr><tr><td align="center">-4</td><td align="center">40</td><td>-4+40=36</td></tr><tr><td align="center">-5</td><td align="center">32</td><td>-5+32=27</td></tr><tr><td align="center">-8</td><td align="center">20</td><td>-8+20=12</td></tr><tr><td align="center">-10</td><td align="center">16</td><td>-10+16=6</td></tr></table>



From this list we can see that 8 and -20 add up to -12 and multiply to -160



Now looking at the expression {{{4x^2-12x-40}}}, replace {{{-12x}}} with {{{8x+-20x}}} (notice {{{8x+-20x}}} adds up to {{{-12x}}}. So it is equivalent to {{{-12x}}})


{{{4x^2+highlight(8x+-20x)+-40}}}



Now let's factor {{{4x^2+8x-20x-40}}} by grouping:



{{{(4x^2+8x)+(-20x-40)}}} Group like terms



{{{4x(x+2)-20(x+2)}}} Factor out the GCF of {{{4x}}} out of the first group. Factor out the GCF of {{{-20}}} out of the second group



{{{(4x-20)(x+2)}}} Since we have a common term of {{{x+2}}}, we can combine like terms


So {{{4x^2+8x-20x-40}}} factors to {{{(4x-20)(x+2)}}}



So this also means that {{{4x^2-12x-40}}} factors to {{{(4x-20)(x+2)}}} (since {{{4x^2-12x-40}}} is equivalent to {{{4x^2+8x-20x-40}}})


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Answer:


So {{{4x^2-12x-40}}} factors to {{{(4x-20)(x+2)}}}




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#2




Looking at {{{2x^2-16x+30}}} we can see that the first term is {{{2x^2}}} and the last term is {{{30}}} where the coefficients are 2 and 30 respectively.


Now multiply the first coefficient 2 and the last coefficient 30 to get 60. Now what two numbers multiply to 60 and add to the  middle coefficient -16? Let's list all of the factors of 60:




Factors of 60:

1,2,3,4,5,6,10,12,15,20,30,60


-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 60

1*60

2*30

3*20

4*15

5*12

6*10

(-1)*(-60)

(-2)*(-30)

(-3)*(-20)

(-4)*(-15)

(-5)*(-12)

(-6)*(-10)


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



Now which of these pairs add to -16? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -16


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">60</td><td>1+60=61</td></tr><tr><td align="center">2</td><td align="center">30</td><td>2+30=32</td></tr><tr><td align="center">3</td><td align="center">20</td><td>3+20=23</td></tr><tr><td align="center">4</td><td align="center">15</td><td>4+15=19</td></tr><tr><td align="center">5</td><td align="center">12</td><td>5+12=17</td></tr><tr><td align="center">6</td><td align="center">10</td><td>6+10=16</td></tr><tr><td align="center">-1</td><td align="center">-60</td><td>-1+(-60)=-61</td></tr><tr><td align="center">-2</td><td align="center">-30</td><td>-2+(-30)=-32</td></tr><tr><td align="center">-3</td><td align="center">-20</td><td>-3+(-20)=-23</td></tr><tr><td align="center">-4</td><td align="center">-15</td><td>-4+(-15)=-19</td></tr><tr><td align="center">-5</td><td align="center">-12</td><td>-5+(-12)=-17</td></tr><tr><td align="center">-6</td><td align="center">-10</td><td>-6+(-10)=-16</td></tr></table>



From this list we can see that -6 and -10 add up to -16 and multiply to 60



Now looking at the expression {{{2x^2-16x+30}}}, replace {{{-16x}}} with {{{-6x+-10x}}} (notice {{{-6x+-10x}}} adds up to {{{-16x}}}. So it is equivalent to {{{-16x}}})


{{{2x^2+highlight(-6x+-10x)+30}}}



Now let's factor {{{2x^2-6x-10x+30}}} by grouping:



{{{(2x^2-6x)+(-10x+30)}}} Group like terms



{{{2x(x-3)-10(x-3)}}} Factor out the GCF of {{{2x}}} out of the first group. Factor out the GCF of {{{-10}}} out of the second group



{{{(2x-10)(x-3)}}} Since we have a common term of {{{x-3}}}, we can combine like terms


So {{{2x^2-6x-10x+30}}} factors to {{{(2x-10)(x-3)}}}



So this also means that {{{2x^2-16x+30}}} factors to {{{(2x-10)(x-3)}}} (since {{{2x^2-16x+30}}} is equivalent to {{{2x^2-6x-10x+30}}})


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Answer:


So {{{2x^2-16x+30}}} factors to {{{(2x-10)(x-3)}}}