SOLUTION: {{{x^2+6x+9}}}. {{{9x^2-6x+1}}}. {{{4x^2+12x+9}}}.

Algebra ->  Expressions-with-variables -> SOLUTION: {{{x^2+6x+9}}}. {{{9x^2-6x+1}}}. {{{4x^2+12x+9}}}.      Log On


   

Question 118891: x%5E2%2B6x%2B9.
9x%5E2-6x%2B1.
4x%5E2%2B12x%2B9.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at x%5E2%2B6x%2B9 we can see that the first term is x%5E2 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

First NumberSecond NumberSum
191+9=10
333+3=6
-1-9-1+(-9)=-10
-3-3-3+(-3)=-6



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


Now looking at the expression x%5E2%2B6x%2B9, replace 6x with 3x%2B3x (notice 3x%2B3x adds up to 6x. So it is equivalent to 6x)

x%5E2%2Bhighlight%283x%2B3x%29%2B9


Now let's factor x%5E2%2B3x%2B3x%2B9 by grouping:


%28x%5E2%2B3x%29%2B%283x%2B9%29 Group like terms


x%28x%2B3%29%2B3%28x%2B3%29 Factor out the GCF of x out of the first group. Factor out the GCF of 3 out of the second group


%28x%2B3%29%28x%2B3%29 Since we have a common term of x%2B3, we can combine like terms

So x%5E2%2B3x%2B3x%2B9 factors to %28x%2B3%29%28x%2B3%29


So this also means that x%5E2%2B6x%2B9 factors to %28x%2B3%29%28x%2B3%29 (since x%5E2%2B6x%2B9 is equivalent to x%5E2%2B3x%2B3x%2B9)


note: %28x%2B3%29%28x%2B3%29 is equivalent to %28x%2B3%29%5E2 since the term x%2B3 occurs twice. So x%5E2%2B6x%2B9 also factors to %28x%2B3%29%5E2


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Answer:

So x%5E2%2B6x%2B9 factors to %28x%2B3%29%5E2








#2



Looking at 9x%5E2-6x%2B1 we can see that the first term is 9x%5E2 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

First NumberSecond NumberSum
191+9=10
333+3=6
-1-9-1+(-9)=-10
-3-3-3+(-3)=-6



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


Now looking at the expression 9x%5E2-6x%2B1, replace -6x with -3x%2B-3x (notice -3x%2B-3x adds up to -6x. So it is equivalent to -6x)

9x%5E2%2Bhighlight%28-3x%2B-3x%29%2B1


Now let's factor 9x%5E2-3x-3x%2B1 by grouping:


%289x%5E2-3x%29%2B%28-3x%2B1%29 Group like terms


3x%283x-1%29-1%283x-1%29 Factor out the GCF of 3x out of the first group. Factor out the GCF of -1 out of the second group


%283x-1%29%283x-1%29 Since we have a common term of 3x-1, we can combine like terms

So 9x%5E2-3x-3x%2B1 factors to %283x-1%29%283x-1%29


So this also means that 9x%5E2-6x%2B1 factors to %283x-1%29%283x-1%29 (since 9x%5E2-6x%2B1 is equivalent to 9x%5E2-3x-3x%2B1)


note: %283x-1%29%283x-1%29 is equivalent to %283x-1%29%5E2 since the term 3x-1 occurs twice. So 9x%5E2-6x%2B1 also factors to %283x-1%29%5E2


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Answer:

So 9x%5E2-6x%2B1 factors to %283x-1%29%5E2