SOLUTION: 9r2-5r-10=0 iam trying to help my son solve this equation (9r-2)(1r-5)

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Question 901893: 9r2-5r-10=0
iam trying to help my son solve this equation
(9r-2)(1r-5)
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
factor
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:



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.



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





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.


complete the square
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:
We have: 1r%5E2%2B-0.555555555555556r%2B-1.11111111111111=0. 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?
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.
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.


The highlighted red part must be equal to -1.11111111111111 (highlighted green part).

-0.555555555555556%5E2%2F4+%2B+othernumber+=+-1.11111111111111, or othernumber+=+-1.11111111111111--0.555555555555556%5E2%2F4+=+-1.18827160493827.
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.

Our equation converted to a square %28r%2B-0.555555555555556%2F2%29%5E2, equated to a number (1.18827160493827).

Since the right part 1.18827160493827 is greater than zero, there are two solutions:


, or





system%28+r+=+1.36785649279714%2C+r+=+-0.812300937241588+%29
Answer: r=1.36785649279714, -0.812300937241588.

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


Starting with the general quadratic


ar%5E2%2Bbr%2Bc=0


the general solution using the quadratic equation is:


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




So lets solve 9%2Ar%5E2-5%2Ar-10=0 ( notice a=9, b=-5, and c=-10)





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




r+=+%285+%2B-+sqrt%28+%28-5%29%5E2-4%2A9%2A-10+%29%29%2F%282%2A9%29 Negate -5 to get 5




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.)




r+=+%285+%2B-+sqrt%28+25%2B360+%29%29%2F%282%2A9%29 Multiply -4%2A-10%2A9 to get 360




r+=+%285+%2B-+sqrt%28+385+%29%29%2F%282%2A9%29 Combine like terms in the radicand (everything under the square root)




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)




r+=+%285+%2B-+sqrt%28385%29%29%2F18 Multiply 2 and 9 to get 18


So now the expression breaks down into two parts


r+=+%285+%2B+sqrt%28385%29%29%2F18 or r+=+%285+-+sqrt%28385%29%29%2F18



Now break up the fraction



r=%2B5%2F18%2Bsqrt%28385%29%2F18 or r=%2B5%2F18-sqrt%28385%29%2F18



Simplify



r=5%2F18%2Bsqrt%28385%29%2F18 or r=5%2F18-sqrt%28385%29%2F18



So the solutions are:

r=5%2F18%2Bsqrt%28385%29%2F18 or r=5%2F18-sqrt%28385%29%2F18



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:

r%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-5%29%5E2-4%2A9%2A-10=385.

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.

r%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+385+%29%29%2F2%5C9+=+1.36785649279714
r%5B2%5D+=+%28-%28-5%29-sqrt%28+385+%29%29%2F2%5C9+=+-0.812300937241588

Quadratic expression 9r%5E2%2B-5r%2B-10 can be factored:
9r%5E2%2B-5r%2B-10+=+9%28r-1.36785649279714%29%2A%28r--0.812300937241588%29
Again, the answer is: 1.36785649279714, -0.812300937241588. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+9%2Ax%5E2%2B-5%2Ax%2B-10+%29