```Question 604356

Looking at the expression {{{5x^2+2x-13}}}, we can see that the first coefficient is {{{5}}}, the second coefficient is {{{2}}}, and the last term is {{{-13}}}.

Now multiply the first coefficient {{{5}}} by the last term {{{-13}}} to get {{{(5)(-13)=-65}}}.

Now the question is: what two whole numbers multiply to {{{-65}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{2}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-65}}} (the previous product).

Factors of {{{-65}}}:

1,5,13,65

-1,-5,-13,-65

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to {{{-65}}}.

1*(-65) = -65
5*(-13) = -65
(-1)*(65) = -65
(-5)*(13) = -65

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{2}}}:

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>-65</font></td><td  align="center"><font color=black>1+(-65)=-64</font></td></tr><tr><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>-13</font></td><td  align="center"><font color=black>5+(-13)=-8</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>65</font></td><td  align="center"><font color=black>-1+65=64</font></td></tr><tr><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>13</font></td><td  align="center"><font color=black>-5+13=8</font></td></tr></table>

From the table, we can see that there are no pairs of numbers which add to {{{2}}}. So {{{5x^2+2x-13}}} cannot be factored.

This tells us that {{{5x^2+2x-13}}} is prime.

This means that you'll have to either complete the square or use the quadratic formula to solve the given equation.```