Algebra help: solving 2x + 0.5 + (-1/x^2) = 0

[GGdub]

I am trying to set the derivative of x^2 + .5x + 1/x set to zero so I can find the optimum x in algebraic terms. The derivative of this problem is 2x+.5+(-1/x^2). I want to set this equal to 0 in terms of x. I am just have trouble with the algebra on this guy. If someone could help me out and try to show the work that would be great. Also I have a TI-89 and I can not figure out how to use the algebra solver. Help on this would be appreciated as well.
Thanks

 

[stapel]

...2x+.5+(-1/x^2). I want to set this equal to 0 in terms of x....

So put it equal to zero:

. . . . .\(\displaystyle 2x\, +\, 0.5\, -\, \frac{1}{x^2}\, =\, 0\)

Multiply through, like you learned back in algebra, to clear the fractions:

. . . . .\(\displaystyle 4x^3\, +\, x^2\, -\, 2\, =\, 0\)

Then use the tools you learned back in algebra, like the Rational Roots Test and synthetic division, to find the zeroes. (A quick graph on the calculator can simplify this process. Check your manual for instructions.)

As it turns out, this has no rational (whole-number or fractional) zeroes, so you'll have to use numerical approximation.

Eliz.

 

[skeeter]

\(\displaystyle f'(x) = 2x + 0.5 - \frac{1}{x^2}\)

since you have to use a calculator anyway, graph the equation of the derivative in your calculator ... look for the single zero.

 

[masters]

After you graph it on your TI-89, looks like the curve crosses the x-axis one time between 0 and 1 (closer to 1).

For the TI-89 and TI-92, you may enter lower bounds or upper bounds from the keyboard instead of arrowing and hitting enter. When it asks for a value, just enter your number and press enter.

Press Math (F5)
Choose Zero (#2)
For the Lower Bound, arrow to the left of the x-intercept and press enter.
For the Upper Bound, arrow to the right of the x-intercept and press enter.
The TI-89 will return a value for x and 0 for y.

Looks to be around .923