Q = sqrt((P^2)/(4 R^10) - (P T)/(R^7)) + (P/R - 2 R^2 T)/(2 R^4) or Q = (P/R - 2 R^2 T)/(2 R^4) - sqrt((P^2)/(4 R^10) - (P T)/(R^7))
Step-by-step explanation:
Solve for Q:
T = sqrt((P Q)/R) - Q R^2
T = sqrt((P Q)/R) - Q R^2 is equivalent to sqrt((P Q)/R) - Q R^2 = T:
sqrt((P Q)/R) - Q R^2 = T
Add Q R^2 to both sides:
sqrt((P Q)/R) = Q R^2 + T
Raise both sides to the power of two:
(P Q)/R = (Q R^2 + T)^2
Expand out terms of the right hand side:
(P Q)/R = Q^2 R^4 + 2 Q R^2 T + T^2
Subtract Q^2 R^4 + 2 Q R^2 T + T^2 from both sides:
(P Q)/R - Q^2 R^4 - 2 Q R^2 T - T^2 = 0
Collect in terms of Q:
-Q^2 R^4 - T^2 + Q (P/R - 2 R^2 T) = 0
Divide both sides by -R^4:
Q^2 + T^2/R^4 - (Q (P/R - 2 R^2 T))/R^4 = 0
Subtract T^2/R^4 from both sides:
Q^2 - (Q (P/R - 2 R^2 T))/R^4 = -T^2/R^4
Add (P/R - 2 R^2 T)^2/(4 R^8) to both sides:
Q^2 - (Q (P/R - 2 R^2 T))/R^4 + (P/R - 2 R^2 T)^2/(4 R^8) = (P/R - 2 R^2 T)^2/(4 R^8) - T^2/R^4
Write the left hand side as a square:
(Q - (P/R - 2 R^2 T)/(2 R^4))^2 = (P/R - 2 R^2 T)^2/(4 R^8) - T^2/R^4
Take the square root of both sides:
Q - (P/R - 2 R^2 T)/(2 R^4) = sqrt((P/R - 2 R^2 T)^2/(4 R^8) - T^2/R^4) or Q - (P/R - 2 R^2 T)/(2 R^4) = -sqrt((P/R - 2 R^2 T)^2/(4 R^8) - T^2/R^4)
Add (P/R - 2 R^2 T)/(2 R^4) to both sides:
Q = (P/R - 2 R^2 T)/(2 R^4) + sqrt(((P/R - 2 R^2 T)^2)/(4 R^8) - (T^2)/(R^4)) or Q - (P/R - 2 R^2 T)/(2 R^4) = -sqrt((P/R - 2 R^2 T)^2/(4 R^8) - T^2/R^4)
(P/R - 2 R^2 T)^2/(4 R^8) - T^2/R^4 = P^2/(4 R^10) - (P T)/R^7:
Q = (P/R - 2 R^2 T)/(2 R^4) + sqrt((P^2)/(4 R^10) - (P T)/(R^7)) or Q - (P/R - 2 R^2 T)/(2 R^4) = -sqrt((P/R - 2 R^2 T)^2/(4 R^8) - T^2/R^4)
Add (P/R - 2 R^2 T)/(2 R^4) to both sides:
Q = sqrt((P^2)/(4 R^10) - (P T)/(R^7)) + (P/R - 2 R^2 T)/(2 R^4) or Q = (P/R - 2 R^2 T)/(2 R^4) - sqrt(((P/R - 2 R^2 T)^2)/(4 R^8) - (T^2)/(R^4))
(P/R - 2 R^2 T)^2/(4 R^8) - T^2/R^4 = P^2/(4 R^10) - (P T)/R^7:
Q = sqrt((P^2)/(4 R^10) - (P T)/(R^7)) + (P/R - 2 R^2 T)/(2 R^4) or Q = (P/R - 2 R^2 T)/(2 R^4) - sqrt((P^2)/(4 R^10) - (P T)/(R^7))
