Question 117062


Looking at {{{5x^2-13x-6}}} we can see that the first term is {{{5x^2}}} and the last term is {{{-6}}} where the coefficients are 5 and -6 respectively.


Now multiply the first coefficient 5 and the last coefficient -6 to get -30. Now what two numbers multiply to -30 and add to the  middle coefficient -13? Let's list all of the factors of -30:




Factors of -30:

1,2,3,5,6,10,15,30


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


These factors pair up and multiply to -30

(1)*(-30)

(2)*(-15)

(3)*(-10)

(5)*(-6)

(-1)*(30)

(-2)*(15)

(-3)*(10)

(-5)*(6)


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



Now which of these pairs add to -13? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -13


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-30</td><td>1+(-30)=-29</td></tr><tr><td align="center">2</td><td align="center">-15</td><td>2+(-15)=-13</td></tr><tr><td align="center">3</td><td align="center">-10</td><td>3+(-10)=-7</td></tr><tr><td align="center">5</td><td align="center">-6</td><td>5+(-6)=-1</td></tr><tr><td align="center">-1</td><td align="center">30</td><td>-1+30=29</td></tr><tr><td align="center">-2</td><td align="center">15</td><td>-2+15=13</td></tr><tr><td align="center">-3</td><td align="center">10</td><td>-3+10=7</td></tr><tr><td align="center">-5</td><td align="center">6</td><td>-5+6=1</td></tr></table>



From this list we can see that 2 and -15 add up to -13 and multiply to -30



Now looking at the expression {{{5x^2-13x-6}}}, replace {{{-13x}}} with {{{2x+-15x}}} (notice {{{2x+-15x}}} adds up to {{{-13x}}}. So it is equivalent to {{{-13x}}})


{{{5x^2+highlight(2x+-15x)+-6}}}



Now let's factor {{{5x^2+2x-15x-6}}} by grouping:



{{{(5x^2+2x)+(-15x-6)}}} Group like terms



{{{x(5x+2)-3(5x+2)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-3}}} out of the second group



{{{(x-3)(5x+2)}}} Since we have a common term of {{{5x+2}}}, we can combine like terms


So {{{5x^2+2x-15x-6}}} factors to {{{(x-3)(5x+2)}}}



So this also means that {{{5x^2-13x-6}}} factors to {{{(x-3)(5x+2)}}} (since {{{5x^2-13x-6}}} is equivalent to {{{5x^2+2x-15x-6}}})


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Answer:


So {{{5x^2-13x-6}}} factors to {{{(x-3)(5x+2)}}}