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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 49 respectively.
Now multiply the first coefficient 1 and the last coefficient 49 to get 49. Now what two numbers multiply to 49 and add to the middle coefficient -14? Let's list all of the factors of 49:
Factors of 49:
1,7
-1,-7 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 49
1*49
7*7
(-1)*(-49)
(-7)*(-7)
note: remember two negative numbers multiplied together make a positive 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
First Number
Second Number
Sum
1
49
1+49=50
7
7
7+7=14
-1
-49
-1+(-49)=-50
-7
-7
-7+(-7)=-14
From this list we can see that -7 and -7 add up to -14 and multiply to 49
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 )
note: is equivalent to since the term occurs twice. So also factors to
(