Question 407530


Looking at {{{1x^2-12x+20}}} we can see that the first term is {{{1x^2}}} and the last term is {{{20}}} where the coefficients are 1 and 20 respectively.


Now multiply the first coefficient 1 and the last coefficient 20 to get 20. Now what two numbers multiply to 20 and add to the  middle coefficient -12? Let's list all of the factors of 20:




Factors of 20:

1,2,4,5,10,20


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


These factors pair up and multiply to 20

1*20

2*10

4*5

(-1)*(-20)

(-2)*(-10)

(-4)*(-5)


note: remember two negative numbers multiplied together make a positive 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">20</td><td>1+20=21</td></tr><tr><td align="center">2</td><td align="center">10</td><td>2+10=12</td></tr><tr><td align="center">4</td><td align="center">5</td><td>4+5=9</td></tr><tr><td align="center">-1</td><td align="center">-20</td><td>-1+(-20)=-21</td></tr><tr><td align="center">-2</td><td align="center">-10</td><td>-2+(-10)=-12</td></tr><tr><td align="center">-4</td><td align="center">-5</td><td>-4+(-5)=-9</td></tr></table>



From this list we can see that -2 and -10 add up to -12 and multiply to 20



Now looking at the expression {{{1x^2-12x+20}}}, replace {{{-12x}}} with {{{-2x+-10x}}} (notice {{{-2x+-10x}}} adds up to {{{-12x}}}. So it is equivalent to {{{-12x}}})


{{{1x^2+highlight(-2x+-10x)+20}}}



Now let's factor {{{1x^2-2x-10x+20}}} by grouping:



{{{(1x^2-2x)+(-10x+20)}}} Group like terms



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



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


So {{{1x^2-2x-10x+20}}} factors to {{{(x-10)(x-2)}}}



So this also means that {{{1x^2-12x+20}}} factors to {{{(x-10)(x-2)}}} (since {{{1x^2-12x+20}}} is equivalent to {{{1x^2-2x-10x+20}}})




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     Answer:

So {{{x^2-12x+20}}} factors to {{{(x-10)(x-2)}}}



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