Question 214986
First, we need to factor {{{-2x^2+13x+7}}}



{{{-2x^2+13x+7}}} Start with the given expression.



{{{-(2x^2-13x-7)}}} Factor out the GCF {{{-1}}}.



Now let's try to factor the inner expression {{{2x^2-13x-7}}}



---------------------------------------------------------------



Looking at the expression {{{2x^2-13x-7}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{-13}}}, and the last term is {{{-7}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{-7}}} to get {{{(2)(-7)=-14}}}.



Now the question is: what two whole numbers multiply to {{{-14}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-13}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-14}}} (the previous product).



Factors of {{{-14}}}:

1,2,7,14

-1,-2,-7,-14



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



These factors pair up and multiply to {{{-14}}}.

1*(-14) = -14
2*(-7) = -14
(-1)*(14) = -14
(-2)*(7) = -14


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-13}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=red>1</font></td><td  align="center"><font color=red>-14</font></td><td  align="center"><font color=red>1+(-14)=-13</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-7</font></td><td  align="center"><font color=black>2+(-7)=-5</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>14</font></td><td  align="center"><font color=black>-1+14=13</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>7</font></td><td  align="center"><font color=black>-2+7=5</font></td></tr></table>



From the table, we can see that the two numbers {{{1}}} and {{{-14}}} add to {{{-13}}} (the middle coefficient).



So the two numbers {{{1}}} and {{{-14}}} both multiply to {{{-14}}} <font size=4><b>and</b></font> add to {{{-13}}}



Now replace the middle term {{{-13x}}} with {{{x-14x}}}. Remember, {{{1}}} and {{{-14}}} add to {{{-13}}}. So this shows us that {{{x-14x=-13x}}}.



{{{2x^2+highlight(x-14x)-7}}} Replace the second term {{{-13x}}} with {{{x-14x}}}.



{{{(2x^2+x)+(-14x-7)}}} Group the terms into two pairs.



{{{x(2x+1)+(-14x-7)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(2x+1)-7(2x+1)}}} Factor out {{{7}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(x-7)(2x+1)}}} Combine like terms. Or factor out the common term {{{2x+1}}}



--------------------------------------------------



So {{{-1(2x^2-13x-7)}}} then factors further to {{{-(x-7)(2x+1)}}}





So {{{-2x^2+13x+7}}} completely factors to {{{-(x-7)(2x+1)}}}.



In other words, {{{-2x^2+13x+7=-(x-7)(2x+1)}}}.



-----------------------------------------------------------



Now let's use the factorization above to solve {{{-2x^2+13x+7=0}}}



{{{-2x^2+13x+7=0}}} Start with the given equation



{{{-(x-7)(2x+1)=0}}} Factor the left side (see above)




Now set each factor equal to zero:

{{{x-7=0}}} or  {{{2x+1=0}}} 



{{{x=7}}} or  {{{x=-1/2}}}    Now solve for x in each case




==================================================================

Answer:



So the solutions are {{{x=7}}} or  {{{x=-1/2}}}