3x^2+9b=(3x+25)(2x+3)
You asked:
Investigate the equation: \(3 {x}^{2} + 9 b = \left(3 x + 25\right) \left(2 x + 3\right)\).
MathBot Answer:
The graph of the equation \(3 {x}^{2} + 9 b = \left(3 x + 25\right) \left(2 x + 3\right)\) is a parabola.
The coordinates of its vertex are: \(\left(- \frac{2581}{108}, - \frac{59}{6}\right)\)
The coordinates of its focus are: \(\left(- \frac{625}{27}, - \frac{59}{6}\right)\)
The directrix has equation: \(\frac{4 b}{3} + \frac{2662}{81} = 0\).
The parabola has parameterization: \[\begin{aligned}b\left(\theta\right) &= - \frac{2581}{108} + \frac{\theta^{2}}{3}\\x\left(\theta\right) &= - \frac{59}{6} + \theta\end{aligned}\]