alg/trig
posted by christina .
9/x + 9/x2= 12

The first thing you want to do is get rid of x in the denominator, because that simplifies things quite a bit. Think of the easiest common denominator between 9/x and 9/(x2), which would be x(x2).
You know that x2/x2 = 1, regardless of the value of x; the same with x/x, so you therefore can multiply 9/x with (x2)/(x2) (which is the same thing as multiplying by one) and 9/(x2) by x/x. This yields
9(x2)/[x(x2)] and 9x/[x(x2)], giving an equation of
9(x2)/[x(x2)] + 9x/[x(x2)] = 12
Because you have a common denominator, you can condense the two terms on the right side of the equation to:
[9(x2) + 9x]/[x(x2)] = 12.
Multiply both sides of the equation by x(x2), eliminating the denominator:
9(x2) + 9x = 12x(x2)
Distribute the terms:
9x18 + 9x = 12x^2  24x
Simplify:
18x18=12x^2  24x
Move all terms to the left side of the equation (subtract 18x from both sides and add 18 to both sides):
0= 12x^2  24x  18x + 18
Simplify:
0=12x^2  42x + 18
You now have a quadratic formula and can now do one of two things: apply the quadratic formula or factor. I'll just factor it out (you can undoubtedly plug the numbers into the quadratic formula on your own time, should you so desire):
0=6(2x1)(x3)
The values that would make this equation true would be
2x1=0 or x3=0, giving xvalues of 0.5 and 3.
Check:
Plug in the xvalues into your equation. Do you get 12?