document.write( "Question 896527: I really need help with this problem!\r
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\n" ); document.write( "\n" ); document.write( "x^2 + 10x + 7 = 0 \n" ); document.write( "
Algebra.Com's Answer #543572 by richwmiller(17219)\"\" \"About 
You can put this solution on YOUR website!
First we try to factor.
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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\":



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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.



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Answer:



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



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


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

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\n" ); document.write( " Starting with the general quadratic
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\n" ); document.write( " \"ax%5E2%2Bbx%2Bc=0\"
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\n" ); document.write( " the general solution using the quadratic equation is:
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\n" ); document.write( " \"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29\"
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\n" ); document.write( " So lets solve \"x%5E2%2B10%2Ax%2B7=0\" ( notice \"a=1\", \"b=10\", and \"c=7\")
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\n" ); document.write( " \"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
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\n" ); document.write( " \"x+=+%28-10+%2B-+sqrt%28+100-4%2A1%2A7+%29%29%2F%282%2A1%29\" Square 10 to get 100
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\n" ); document.write( " \"x+=+%28-10+%2B-+sqrt%28+100%2B-28+%29%29%2F%282%2A1%29\" Multiply \"-4%2A7%2A1\" to get \"-28\"
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\n" ); document.write( " \"x+=+%28-10+%2B-+sqrt%28+72+%29%29%2F%282%2A1%29\" Combine like terms in the radicand (everything under the square root)
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\n" ); document.write( " \"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)
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\n" ); document.write( " \"x+=+%28-10+%2B-+6%2Asqrt%282%29%29%2F2\" Multiply 2 and 1 to get 2
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\n" ); document.write( " So now the expression breaks down into two parts
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\n" ); document.write( " \"x+=+%28-10+%2B+6%2Asqrt%282%29%29%2F2\" or \"x+=+%28-10+-+6%2Asqrt%282%29%29%2F2\"
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\n" ); document.write( " Now break up the fraction
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\n" ); document.write( " \"x=-10%2F2%2B6%2Asqrt%282%29%2F2\" or \"x=-10%2F2-6%2Asqrt%282%29%2F2\"
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\n" ); document.write( " Simplify
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\n" ); document.write( " \"x=-5%2B3%2Asqrt%282%29\" or \"x=-5-3%2Asqrt%282%29\"
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\n" ); document.write( " So the solutions are:
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\n" ); document.write( " \"x=-5%2B3%2Asqrt%282%29\" or \"x=-5-3%2Asqrt%282%29\"
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