I posted the question previously, but I had transcribed part of the formula incorrectly, hence the answer I received from fedja (shown below) was correct, but the question was wrong. I've tried to rework his answer but so far haven't managed it. For friction loss, I previously had Q/100 doubled but it should be squared.
In dealing with fire ground math for a smooth bore tip and friction loss, there are two formulas. One that calculates the flow (gallons per minute) and one that determines the friction loss based on the hose. The sum of those answers is the pressure that the fire pumper will pump at. I want to be able to determine the flow and friction loss given a pump pressure.
The formula for flow is: $$GPM=29.71\times D^2\times\sqrt{np}\;\;where\,D\,is\,nozzle\, diameter\;and\,np\,is \;nozzle \;pressure$$ and for friction loss is: $$FL=C\times \color{blue}{ \left( \frac{Q}{100} \right)^2 } \times\frac{L}{100}\;\;where\,C\,is\,hose\,friction\,loss\,coefficient,\;Q\,is\, GPM,\,and\,L\,is\,length $$ So, if D is 1.125, np is 50, then GPM = 266
With a C of 8 and L of 100 Then FL= 43
Resulting pump pressure is 50 + 43 = 93
So, what I want to be able to do is calculate gallons per minute with a pump pressure of 93? (Given in this case the same parameters for D, C, and L).
Fedja's answer, that worked great for what I asked was: $A=\frac 1{29.71 D^2}$ and $B=\frac{2CL}{100^2}$ (and assuming that $Q$ and $GPM$ are exactly the same thing), you get the equation $A^2\times GPM^2+ B\times GPM=pump\ pressure=PP$ so, by the quadratic formula, $$ GPM=\frac{-B+\sqrt{B^2+4\times PP\times A^2}}{2A^2} $$ How do I modify this for the correct formula for friction loss?
Ok, with @fedja's encouragement I took another stab and believe I have this figured out. I started with some easy numbers to work with and then tested some real number situations.
So $A=\frac 1{29.71 D^2}\enspace$ and $\enspace B=\frac{CL}{100\times100^2}$
$GPM \times A = \sqrt{NP}\enspace therefore \enspace GPM^2 \times A^2 = NP\enspace$and $\enspace GPM^2 \times B = FL$
$PP = FL + NP$, so:
$(GPM^2 \times A^2)+(GPM^2\times B)=FL+NP=PP$
$GPM^2 \times (A^2+B)=PP\enspace$ leading to:
$GPM = \sqrt{ \left(\frac{PP}{A^2 + B} \right)} $