How to calculate the 3dB point?

I understand this until the final two lines. How does he get from 27.8766 to 1149.8hz and 7224.7rad/s?

The answer to your first question comes from a little algebra. Subtract 7 from both sides, divide by 1.579e-5, and take the square root:

$$\sqrt{\frac{27.8766 - 7}{1.579\times10^{-5}}} = 1149.8$$

The conversion to radians per second is done by multiplying by $$\2\pi\$$ as they show.

To answer your follow up question about where $$\1.579\times10^{-5}\$$ comes from, it is equal to the expression from earlier in the same line, after factoring $$\(2\pi f_{6dB})^2\$$ into $$\(2\pi)^2(f_{6dB})^2\$$:

$$(4\times10^{-7})\cdot(2\pi)^2$$

• Thanks for the reply, I'm also a little confused as to where the 1.579x10^5 comes from? – James Apr 30 at 18:01
• @James -- See edit – Justin Apr 30 at 18:05
• @James I think your visual cortex might have missed the arrow in the middle of the third last line where it jumps to a whole new equation that does not equate to F(f_6db). It's a whole new inequality. I missed it too the first few times I looked at it. – DKNguyen Apr 30 at 18:28