document.write( "Question 901893: 9r2-5r-10=0
\n" ); document.write( "iam trying to help my son solve this equation
\n" ); document.write( "(9r-2)(1r-5) \n" ); document.write( "
Algebra.Com's Answer #547010 by richwmiller(17219)\"\" \"About 
You can put this solution on YOUR website!
factor
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression \"9r%5E2-5r-10\", we can see that the first coefficient is \"9\", the second coefficient is \"-5\", and the last term is \"-10\".



Now multiply the first coefficient \"9\" by the last term \"-10\" to get \"%289%29%28-10%29=-90\".



Now the question is: what two whole numbers multiply to \"-90\" (the previous product) and add to the second coefficient \"-5\"?



To find these two numbers, we need to list all of the factors of \"-90\" (the previous product).



Factors of \"-90\":

1,2,3,5,6,9,10,15,18,30,45,90

-1,-2,-3,-5,-6,-9,-10,-15,-18,-30,-45,-90



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



These factors pair up and multiply to \"-90\".

1*(-90) = -90
2*(-45) = -90
3*(-30) = -90
5*(-18) = -90
6*(-15) = -90
9*(-10) = -90
(-1)*(90) = -90
(-2)*(45) = -90
(-3)*(30) = -90
(-5)*(18) = -90
(-6)*(15) = -90
(-9)*(10) = -90


Now let's add up each pair of factors to see if one pair adds to the middle coefficient \"-5\":



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First NumberSecond NumberSum
1-901+(-90)=-89
2-452+(-45)=-43
3-303+(-30)=-27
5-185+(-18)=-13
6-156+(-15)=-9
9-109+(-10)=-1
-190-1+90=89
-245-2+45=43
-330-3+30=27
-518-5+18=13
-615-6+15=9
-910-9+10=1




From the table, we can see that there are no pairs of numbers which add to \"-5\". So \"9r%5E2-5r-10\" cannot be factored.



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Answer:



So \"9%2Ar%5E2-5%2Ar-10\" doesn't factor at all (over the rational numbers).



So \"9%2Ar%5E2-5%2Ar-10\" is prime.


\n" ); document.write( "\n" ); document.write( "complete the square
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Solved by pluggable solver: COMPLETING THE SQUARE solver for quadratics
Read this lesson on completing the square by prince_abubu, if you do not know how to complete the square.
Let's convert \"9r%5E2%2B-5r%2B-10=0\" to standard form by dividing both sides by 9:
\n" ); document.write( "We have: \"1r%5E2%2B-0.555555555555556r%2B-1.11111111111111=0\". \n" ); document.write( "What we want to do now is to change this equation to a complete square \"%28r%2Bsomenumber%29%5E2+%2B+othernumber\". How can we find out values of somenumber and othernumber that would make it work?
\n" ); document.write( "Look at \"%28r%2Bsomenumber%29%5E2\": \"%28r%2Bsomenumber%29%5E2+=+r%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2\". Since the coefficient in our equation \"1r%5E2%2Bhighlight_red%28+-0.555555555555556%29+%2A+r%2B-1.11111111111111=0\" that goes in front of r is -0.555555555555556, we know that -0.555555555555556=2*somenumber, or \"somenumber+=+-0.555555555555556%2F2\". So, we know that our equation can be rewritten as \"%28r%2B-0.555555555555556%2F2%29%5E2+%2B+othernumber\", and we do not yet know the other number.
\n" ); document.write( "We are almost there. Finding the other number is simply a matter of not making too many mistakes. We need to find 'other number' such that \"%28r%2B-0.555555555555556%2F2%29%5E2+%2B+othernumber\" is equivalent to our original equation \"1r%5E2%2B-0.555555555555556r%2Bhighlight_green%28+-1.11111111111111+%29=0\".
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\n" ); document.write( " The highlighted red part must be equal to -1.11111111111111 (highlighted green part).
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\n" ); document.write( " \"-0.555555555555556%5E2%2F4+%2B+othernumber+=+-1.11111111111111\", or \"othernumber+=+-1.11111111111111--0.555555555555556%5E2%2F4+=+-1.18827160493827\".
\n" ); document.write( "So, the equation converts to \"%28r%2B-0.555555555555556%2F2%29%5E2+%2B+-1.18827160493827+=+0\", or \"%28r%2B-0.555555555555556%2F2%29%5E2+=+1.18827160493827\".
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\n" ); document.write( " Our equation converted to a square \"%28r%2B-0.555555555555556%2F2%29%5E2\", equated to a number (1.18827160493827).
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\n" ); document.write( " Since the right part 1.18827160493827 is greater than zero, there are two solutions:
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\n" ); document.write( " \"system%28+r+=+1.36785649279714%2C+r+=+-0.812300937241588+%29\"
\n" ); document.write( "Answer: r=1.36785649279714, -0.812300937241588.\n" ); document.write( "
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Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for r:

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\n" ); document.write( " Starting with the general quadratic
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\n" ); document.write( " \"ar%5E2%2Bbr%2Bc=0\"
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\n" ); document.write( " the general solution using the quadratic equation is:
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\n" ); document.write( " \"r+=+%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 \"9%2Ar%5E2-5%2Ar-10=0\" ( notice \"a=9\", \"b=-5\", and \"c=-10\")
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\n" ); document.write( " \"r+=+%28--5+%2B-+sqrt%28+%28-5%29%5E2-4%2A9%2A-10+%29%29%2F%282%2A9%29\" Plug in a=9, b=-5, and c=-10
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\n" ); document.write( " \"r+=+%285+%2B-+sqrt%28+%28-5%29%5E2-4%2A9%2A-10+%29%29%2F%282%2A9%29\" Negate -5 to get 5
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\n" ); document.write( " \"r+=+%285+%2B-+sqrt%28+25-4%2A9%2A-10+%29%29%2F%282%2A9%29\" Square -5 to get 25 (note: remember when you square -5, you must square the negative as well. This is because \"%28-5%29%5E2=-5%2A-5=25\".)
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\n" ); document.write( " \"r+=+%285+%2B-+sqrt%28+25%2B360+%29%29%2F%282%2A9%29\" Multiply \"-4%2A-10%2A9\" to get \"360\"
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\n" ); document.write( " \"r+=+%285+%2B-+sqrt%28+385+%29%29%2F%282%2A9%29\" Combine like terms in the radicand (everything under the square root)
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\n" ); document.write( " \"r+=+%285+%2B-+sqrt%28385%29%29%2F%282%2A9%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( " \"r+=+%285+%2B-+sqrt%28385%29%29%2F18\" Multiply 2 and 9 to get 18
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\n" ); document.write( " So now the expression breaks down into two parts
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\n" ); document.write( " \"r+=+%285+%2B+sqrt%28385%29%29%2F18\" or \"r+=+%285+-+sqrt%28385%29%29%2F18\"
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\n" ); document.write( " Now break up the fraction
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\n" ); document.write( " \"r=%2B5%2F18%2Bsqrt%28385%29%2F18\" or \"r=%2B5%2F18-sqrt%28385%29%2F18\"
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\n" ); document.write( " Simplify
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\n" ); document.write( " \"r=5%2F18%2Bsqrt%28385%29%2F18\" or \"r=5%2F18-sqrt%28385%29%2F18\"
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\n" ); document.write( " So the solutions are:
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\n" ); document.write( " \"r=5%2F18%2Bsqrt%28385%29%2F18\" or \"r=5%2F18-sqrt%28385%29%2F18\"
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Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation \"ar%5E2%2Bbr%2Bc=0\" (in our case \"9r%5E2%2B-5r%2B-10+=+0\") has the following solutons:
\n" ); document.write( "
\n" ); document.write( " \"r%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"
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\n" ); document.write( " For these solutions to exist, the discriminant \"b%5E2-4ac\" should not be a negative number.
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\n" ); document.write( " First, we need to compute the discriminant \"b%5E2-4ac\": \"b%5E2-4ac=%28-5%29%5E2-4%2A9%2A-10=385\".
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\n" ); document.write( " Discriminant d=385 is greater than zero. That means that there are two solutions: \"+x%5B12%5D+=+%28--5%2B-sqrt%28+385+%29%29%2F2%5Ca\".
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\n" ); document.write( " \"r%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+385+%29%29%2F2%5C9+=+1.36785649279714\"
\n" ); document.write( " \"r%5B2%5D+=+%28-%28-5%29-sqrt%28+385+%29%29%2F2%5C9+=+-0.812300937241588\"
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\n" ); document.write( " Quadratic expression \"9r%5E2%2B-5r%2B-10\" can be factored:
\n" ); document.write( " \"9r%5E2%2B-5r%2B-10+=+9%28r-1.36785649279714%29%2A%28r--0.812300937241588%29\"
\n" ); document.write( " Again, the answer is: 1.36785649279714, -0.812300937241588.\n" ); document.write( "Here's your graph:
\n" ); document.write( "\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+9%2Ax%5E2%2B-5%2Ax%2B-10+%29\"
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