Dancers all over

In $\color{#D61F06}{Bingo}$ 's equation, put the value of $y$ from $\color{#3D99F6}{Mingo}$ 's equation. Then you get the equation

$x+44 = x^3 -52x+96$

That is , $x^3-53x + 52=0$

By observation only, $1$ is a root of this equation, thus 1 is a common solution of $\color{#D61F06}{Bingo}$ and $\color{#3D99F6}{Mingo}$ 's equations for the point of yelling.

And the equations actually are

$\color{#D61F06}{Bingo} :::: \color{#D61F06}{y= (x-2)(x-6)(x+8)}$

$\color{#3D99F6}{Mingo} :::: \color{#3D99F6}{ y= x+44}$

$\color{#20A900}{Tingo} :::: \color{#20A900}{y = (x-16) (x-4) (x-2)^2}$

Even if you can't factorize, just try putting $x=1$ in all equations, everytime you get the value of $y$ as $45$ .

Hence the point where they meet is $(1,45)$ and hence asked $\color{#69047E}{answer}$ is $\color{#69047E}{\Huge{\boxed{46}}}$

Intersecting Bingo and Mingo, we have $x^3-52x+96=x+44\implies x=1,\frac{-1\pm\sqrt{209}}{2}$ . Hence, $x=1$ because the coordinates must be integers. All that is left is to check that $y=45$ in all cases, which it does.