document.write( "Question 464645: How do you factor x^2+2x-276=0? \n" ); document.write( "

Algebra.Com's Answer #318293 by MathLover1(20850) $\"\"$ $\"About$

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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"x%5E2%2B2x-276\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"2\"$ , and the last term is $\"-276\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-276\"$ to get $\"%281%29%28-276%29=-276\"$ .

Now the question is: what two whole numbers multiply to $\"-276\"$ (the previous product) and add to the second coefficient $\"2\"$ ?

To find these two numbers, we need to list all of the factors of $\"-276\"$ (the previous product).

Factors of $\"-276\"$ :

1,2,3,4,6,12,23,46,69,92,138,276

-1,-2,-3,-4,-6,-12,-23,-46,-69,-92,-138,-276

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to $\"-276\"$ .

1*(-276) = -276
2*(-138) = -276
3*(-92) = -276
4*(-69) = -276
6*(-46) = -276
12*(-23) = -276
(-1)*(276) = -276
(-2)*(138) = -276
(-3)*(92) = -276
(-4)*(69) = -276
(-6)*(46) = -276
(-12)*(23) = -276

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"2\"$ :

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First Number	Second Number	Sum
1	-276	1+(-276)=-275
2	-138	2+(-138)=-136
3	-92	3+(-92)=-89
4	-69	4+(-69)=-65
6	-46	6+(-46)=-40
12	-23	12+(-23)=-11
-1	276	-1+276=275
-2	138	-2+138=136
-3	92	-3+92=89
-4	69	-4+69=65
-6	46	-6+46=40
-12	23	-12+23=11

From the table, we can see that there are no pairs of numbers which add to $\"2\"$ . So $\"x%5E2%2B2x-276\"$ cannot be factored.

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Answer:

So $\"x%5E2%2B2%2Ax-276\"$ doesn't factor at all (over the rational numbers).

So $\"x%5E2%2B2%2Ax-276\"$ is prime.

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