Question 195604


Looking at {{{2x^2+x-6}}} we can see that the first term is {{{2x^2}}} and the last term is {{{-6}}} where the coefficients are 2 and -6 respectively.


Now multiply the first coefficient 2 and the last coefficient -6 to get -12. Now what two numbers multiply to -12 and add to the  middle coefficient 1? Let's list all of the factors of -12:




Factors of -12:

1,2,3,4,6,12


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


These factors pair up and multiply to -12

(1)*(-12)

(2)*(-6)

(3)*(-4)

(-1)*(12)

(-2)*(6)

(-3)*(4)


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



Now which of these pairs add to 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-12</td><td>1+(-12)=-11</td></tr><tr><td align="center">2</td><td align="center">-6</td><td>2+(-6)=-4</td></tr><tr><td align="center">3</td><td align="center">-4</td><td>3+(-4)=-1</td></tr><tr><td align="center">-1</td><td align="center">12</td><td>-1+12=11</td></tr><tr><td align="center">-2</td><td align="center">6</td><td>-2+6=4</td></tr><tr><td align="center">-3</td><td align="center">4</td><td>-3+4=1</td></tr></table>



From this list we can see that -3 and 4 add up to 1 and multiply to -12



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


{{{2x^2+highlight(-3x+4x)-6}}}



Now let's factor {{{2x^2-3x+4x-6}}} by grouping:



{{{(2x^2-3x)+(4x-6)}}} Group like terms



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



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


So {{{2x^2-3x+4x-6}}} factors to {{{(x+2)(2x-3)}}}



So this also means that {{{2x^2+x-6}}} factors to {{{(x+2)(2x-3)}}} (since {{{2x^2+x-6}}} is equivalent to {{{2x^2-3x+4x-6}}})




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     Answer:

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