Frequency problem - Physics 152 (or Phys 2) question

Question

A 101-Ω resistor,a 0.234-µF capacitor, and a 5.69-mH inductor are connected in series to a generator whose voltage is 22.4 V. The current in the circuit is 0.128 A. Because of the shape of the current-versus-frequency graph:

there are two possible values for the frequency that correspond to this current. Obtain these two values. Give the label (a) to the smaller value and the label (b) to the larger value. Note: The ac current and voltage are rms values and power is an average value unless indicated otherwise.

Answer 1

Since the current is given, Z can be calculated from Z = E/I, then the value of Z can be substituted in the equation below that will look like this:

Z^2 = R^2 +(X_L – X_C)^2 where X_L = 2πfL and X_{C =} 1/2πfC)

There is only one unknown in the equation above, which is the frequency, f. This equation can be expanded and solved for f, resulting in a quadratic equation in f. 

As quadratic equations have two roots, two frequencies will result, which answers the original question.

I will leave the calculation to you.