Question 139520

Looking at {{{49x^2-14x-3}}} we can see that the first term is {{{49x^2}}} and the last term is {{{-3}}} where the coefficients are 49 and -3 respectively.


Now multiply the first coefficient 49 and the last coefficient -3 to get -147. Now what two numbers multiply to -147 and add to the  middle coefficient -14? Let's list all of the factors of -147:




Factors of -147:

1,3,7,21,49,147


-1,-3,-7,-21,-49,-147 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -147

(1)*(-147)

(3)*(-49)

(7)*(-21)

(-1)*(147)

(-3)*(49)

(-7)*(21)


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



Now which of these pairs add to -14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -14


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-147</td><td>1+(-147)=-146</td></tr><tr><td align="center">3</td><td align="center">-49</td><td>3+(-49)=-46</td></tr><tr><td align="center">7</td><td align="center">-21</td><td>7+(-21)=-14</td></tr><tr><td align="center">-1</td><td align="center">147</td><td>-1+147=146</td></tr><tr><td align="center">-3</td><td align="center">49</td><td>-3+49=46</td></tr><tr><td align="center">-7</td><td align="center">21</td><td>-7+21=14</td></tr></table>



From this list we can see that 7 and -21 add up to -14 and multiply to -147



Now looking at the expression {{{49x^2-14x-3}}}, replace {{{-14x}}} with {{{7x+-21x}}} (notice {{{7x+-21x}}} adds up to {{{-14x}}}. So it is equivalent to {{{-14x}}})


{{{49x^2+highlight(7x+-21x)+-3}}}



Now let's factor {{{49x^2+7x-21x-3}}} by grouping:



{{{(49x^2+7x)+(-21x-3)}}} Group like terms



{{{7x(7x+1)-3(7x+1)}}} Factor out the GCF of {{{7x}}} out of the first group. Factor out the GCF of {{{-3}}} out of the second group



{{{(7x-3)(7x+1)}}} Since we have a common term of {{{7x+1}}}, we can combine like terms


So {{{49x^2+7x-21x-3}}} factors to {{{(7x-3)(7x+1)}}}



So this also means that {{{49x^2-14x-3}}} factors to {{{(7x-3)(7x+1)}}} (since {{{49x^2-14x-3}}} is equivalent to {{{49x^2+7x-21x-3}}})






{{{(7x-3)(7x+1)=0}}} Set the factorization equal to zero




Now set each factor equal to zero:

{{{7x-3=0}}} or  {{{7x+1=0}}} 


{{{x=3/7}}} or  {{{x=-1/7}}}    Now solve for x in each case




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Answer:


So our solutions are


 {{{x=3/7}}} or  {{{x=-1/7}}}