Direct Solution for Reinforcement Ratio

[Jpr1216]

In the Structural Engineering Reference Manual they give a direct solution for solving for rho for beams with tension reinforcing but they do not derive the formula. This seems like a pretty basic step that I should know but I'm having trouble with the math. I think they're using the quadratic formula but I can't seem to figure out how they simplify the formula as shown in the attached image file. Kudos to anyone who can show me the solution! Thanks.

(PS you can ignore the red pen writing)

 

[Ingenuity]

See the following:

Bit difficult to read the above, so see attached file for a PDF copy.

 

[Jpr1216]

Well done! Now, did you figure that out on your own or did you find it in a book?? Either way, thanks! Appreciate it very much.

 

[Ingenuity]

You are welcome.

No textbooks required - and the last time I derived it was more than 30 years ago.

This is the simple case.

I challenge you to consider a non-rectangular section with top and bottom layers of rebar with prestressing.