Bernoulli's Theorem in fluid mechanics:
For the steady, incompressible, frictionless (inviscid), and irrotational flow of a fluid along a streamline, total mechanical energy per unit weight is conserved:

\[\frac{p}{\rho g} + \frac{v^2}{2g} + z = \text{Constant}\]

where:

• \(\frac{p}{\rho g}\) = Pressure head
• \(\frac{v^2}{2g}\) = Velocity head - \(z\) = Elevation (potential) head

Nature of roots of quadratic equation

\(aX^2 + bX + c = 0\): Discriminant \(D = b^2 - 4ac\) For \(X^2 - 6X + 9 = 0\): \(a = 1\), \(b = -6\), \(c = 9\) \[D = (-6)^2 - 4(1)(9) = 36 - 36 = 0\]

When

\(D = 0\), the roots are \lt strong\gt real, rational, and equal\lt /strong\gt : \[X = \frac{-(-6) \pm 0}{2(1)} = 3, 3\]

Most economical / efficient trapezoidal channel section:
For a given cross-sectional area and bed slope, discharge is maximized when the hydraulic radius $R = A/P$ is maximum (wetted perimeter $P$ is minimum).
The geometric condition for the most economical trapezoidal section requires that the side slopes and bottom are tangent to a semi-circle described on the water surface.
When $\theta$ is the angle of inclination of the side slope with the horizontal channel bed, the bed width is related to flow depth by:

$$b = 2d \tan(\theta/2)$$

Under this condition, the top width $T = 2 \times \text{sloping side}$, and $R = d/2$.