SOLUTION: factor x^2+7x+10=0

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Question 145612: factor x^2+7x+10=0
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Looking at we can see that the first term is and the last term is where the coefficients are 1 and 10 respectively.

Now multiply the first coefficient 1 and the last coefficient 10 to get 10. Now what two numbers multiply to 10 and add to the middle coefficient 7? Let's list all of the factors of 10:

Factors of 10:
1,2,5,10

-1,-2,-5,-10 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 10
1*10
2*5
(-1)*(-10)
(-2)*(-5)

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

Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7

First NumberSecond NumberSum
1101+10=11
252+5=7
-1-10-1+(-10)=-11
-2-5-2+(-5)=-7

From this list we can see that 2 and 5 add up to 7 and multiply to 10

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

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