SOLUTION: I really need help with this problem! x^2 + 10x + 7 = 0

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Question 896527: I really need help with this problem!

x^2 + 10x + 7 = 0
Found 2 solutions by JulietG, richwmiller:
Answer by JulietG(1812) About Me  (Show Source):
You can put this solution on YOUR website!
Are you sure you've written the problem correctly?
There is no solution with rational numbers as written.
However, if it's x^2+7x+10, then it's simple factoring
(x+5)(x+2) = x^2+7x+10
X = -5,-2

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
First we try to factor.
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2%2B10x%2B7, we can see that the first coefficient is 1, the second coefficient is 10, and the last term is 7.



Now multiply the first coefficient 1 by the last term 7 to get %281%29%287%29=7.



Now the question is: what two whole numbers multiply to 7 (the previous product) and add to the second coefficient 10?



To find these two numbers, we need to list all of the factors of 7 (the previous product).



Factors of 7:

1,7

-1,-7



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



These factors pair up and multiply to 7.

1*7 = 7
(-1)*(-7) = 7


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 10:



First NumberSecond NumberSum
171+7=8
-1-7-1+(-7)=-8




From the table, we can see that there are no pairs of numbers which add to 10. So x%5E2%2B10x%2B7 cannot be factored.



===============================================================





Answer:



So x%5E2%2B10%2Ax%2B7 doesn't factor at all (over the rational numbers).



So x%5E2%2B10%2Ax%2B7 is prime.


then we use the quadratic_formula
Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve x%5E2%2B10%2Ax%2B7=0 ( notice a=1, b=10, and c=7)





x+=+%28-10+%2B-+sqrt%28+%2810%29%5E2-4%2A1%2A7+%29%29%2F%282%2A1%29 Plug in a=1, b=10, and c=7




x+=+%28-10+%2B-+sqrt%28+100-4%2A1%2A7+%29%29%2F%282%2A1%29 Square 10 to get 100




x+=+%28-10+%2B-+sqrt%28+100%2B-28+%29%29%2F%282%2A1%29 Multiply -4%2A7%2A1 to get -28




x+=+%28-10+%2B-+sqrt%28+72+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-10+%2B-+6%2Asqrt%282%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-10+%2B-+6%2Asqrt%282%29%29%2F2 Multiply 2 and 1 to get 2


So now the expression breaks down into two parts


x+=+%28-10+%2B+6%2Asqrt%282%29%29%2F2 or x+=+%28-10+-+6%2Asqrt%282%29%29%2F2



Now break up the fraction



x=-10%2F2%2B6%2Asqrt%282%29%2F2 or x=-10%2F2-6%2Asqrt%282%29%2F2



Simplify



x=-5%2B3%2Asqrt%282%29 or x=-5-3%2Asqrt%282%29



So the solutions are:

x=-5%2B3%2Asqrt%282%29 or x=-5-3%2Asqrt%282%29