Question 118891


Looking at {{{x^2+6x+9}}} we can see that the first term is {{{x^2}}} and the last term is {{{9}}} where the coefficients are 1 and 9 respectively.


Now multiply the first coefficient 1 and the last coefficient 9 to get 9. Now what two numbers multiply to 9 and add to the  middle coefficient 6? Let's list all of the factors of 9:




Factors of 9:

1,3


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


These factors pair up and multiply to 9

1*9

3*3

(-1)*(-9)

(-3)*(-3)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to 6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 6


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">9</td><td>1+9=10</td></tr><tr><td align="center">3</td><td align="center">3</td><td>3+3=6</td></tr><tr><td align="center">-1</td><td align="center">-9</td><td>-1+(-9)=-10</td></tr><tr><td align="center">-3</td><td align="center">-3</td><td>-3+(-3)=-6</td></tr></table>



From this list we can see that 3 and 3 add up to 6 and multiply to 9



Now looking at the expression {{{x^2+6x+9}}}, replace {{{6x}}} with {{{3x+3x}}} (notice {{{3x+3x}}} adds up to {{{6x}}}. So it is equivalent to {{{6x}}})


{{{x^2+highlight(3x+3x)+9}}}



Now let's factor {{{x^2+3x+3x+9}}} by grouping:



{{{(x^2+3x)+(3x+9)}}} Group like terms



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



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


So {{{x^2+3x+3x+9}}} factors to {{{(x+3)(x+3)}}}



So this also means that {{{x^2+6x+9}}} factors to {{{(x+3)(x+3)}}} (since {{{x^2+6x+9}}} is equivalent to {{{x^2+3x+3x+9}}})



note:  {{{(x+3)(x+3)}}} is equivalent to  {{{(x+3)^2}}} since the term {{{x+3}}} occurs twice. So {{{x^2+6x+9}}} also factors to {{{(x+3)^2}}}



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Answer:


So {{{x^2+6x+9}}} factors to {{{(x+3)^2}}} 





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#2




Looking at {{{9x^2-6x+1}}} we can see that the first term is {{{9x^2}}} and the last term is {{{1}}} where the coefficients are 9 and 1 respectively.


Now multiply the first coefficient 9 and the last coefficient 1 to get 9. Now what two numbers multiply to 9 and add to the  middle coefficient -6? Let's list all of the factors of 9:




Factors of 9:

1,3


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


These factors pair up and multiply to 9

1*9

3*3

(-1)*(-9)

(-3)*(-3)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to -6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -6


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">9</td><td>1+9=10</td></tr><tr><td align="center">3</td><td align="center">3</td><td>3+3=6</td></tr><tr><td align="center">-1</td><td align="center">-9</td><td>-1+(-9)=-10</td></tr><tr><td align="center">-3</td><td align="center">-3</td><td>-3+(-3)=-6</td></tr></table>



From this list we can see that -3 and -3 add up to -6 and multiply to 9



Now looking at the expression {{{9x^2-6x+1}}}, replace {{{-6x}}} with {{{-3x+-3x}}} (notice {{{-3x+-3x}}} adds up to {{{-6x}}}. So it is equivalent to {{{-6x}}})


{{{9x^2+highlight(-3x+-3x)+1}}}



Now let's factor {{{9x^2-3x-3x+1}}} by grouping:



{{{(9x^2-3x)+(-3x+1)}}} Group like terms



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



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


So {{{9x^2-3x-3x+1}}} factors to {{{(3x-1)(3x-1)}}}



So this also means that {{{9x^2-6x+1}}} factors to {{{(3x-1)(3x-1)}}} (since {{{9x^2-6x+1}}} is equivalent to {{{9x^2-3x-3x+1}}})



note:  {{{(3x-1)(3x-1)}}} is equivalent to  {{{(3x-1)^2}}} since the term {{{3x-1}}} occurs twice. So {{{9x^2-6x+1}}} also factors to {{{(3x-1)^2}}}



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Answer:


So {{{9x^2-6x+1}}} factors to {{{(3x-1)^2}}}