SOLUTION: Find the real or imaginary numbers of the equation: x^-2 - 9x^-1 + 18=0

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Question 97352: Find the real or imaginary numbers of the equation:
x^-2 - 9x^-1 + 18=0
Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E-2+-+9x%5E-1+%2B+18=0



1%2Fx%5E2+-+9%2Fx+%2B+18=0 Flip all the variables with negative exponents


x%5E2%281%2Fx%5E2+-+9%2Fx+%2B+18%29=x%5E2%2A0 Multiply both sides by the LCD


1+-+9x+%2B+18x%5E2=0 Distribute and multiply


18x%5E2-+9x+%2B1+=0 Rearrange the terms




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 18%2Ax%5E2-9%2Ax%2B1=0 ( notice a=18, b=-9, and c=1)

x+=+%28--9+%2B-+sqrt%28+%28-9%29%5E2-4%2A18%2A1+%29%29%2F%282%2A18%29 Plug in a=18, b=-9, and c=1



x+=+%289+%2B-+sqrt%28+%28-9%29%5E2-4%2A18%2A1+%29%29%2F%282%2A18%29 Negate -9 to get 9



x+=+%289+%2B-+sqrt%28+81-4%2A18%2A1+%29%29%2F%282%2A18%29 Square -9 to get 81 (note: remember when you square -9, you must square the negative as well. This is because %28-9%29%5E2=-9%2A-9=81.)



x+=+%289+%2B-+sqrt%28+81%2B-72+%29%29%2F%282%2A18%29 Multiply -4%2A1%2A18 to get -72



x+=+%289+%2B-+sqrt%28+9+%29%29%2F%282%2A18%29 Combine like terms in the radicand (everything under the square root)



x+=+%289+%2B-+3%29%2F%282%2A18%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%289+%2B-+3%29%2F36 Multiply 2 and 18 to get 36

So now the expression breaks down into two parts

x+=+%289+%2B+3%29%2F36 or x+=+%289+-+3%29%2F36

Lets look at the first part:

x=%289+%2B+3%29%2F36

x=12%2F36 Add the terms in the numerator
x=1%2F3 Divide

So one answer is
x=1%2F3



Now lets look at the second part:

x=%289+-+3%29%2F36

x=6%2F36 Subtract the terms in the numerator
x=1%2F6 Divide

So another answer is
x=1%2F6

So our solutions are:
x=1%2F3 or x=1%2F6

Notice when we graph 18%2Ax%5E2-9%2Ax%2B1, we get:

+graph%28+500%2C+500%2C+-9%2C+11%2C+-9%2C+11%2C18%2Ax%5E2%2B-9%2Ax%2B1%29+

and we can see that the roots are x=1%2F3 and x=1%2F6. This verifies our answer

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
x^-2 - 9x^-1 + 18=0
(x^-1-3)(x^-1-6)=0
x^-1= 3 or x^-1=6
x=1/3 or x=1/6
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Ceers,
Stan H.