Given 3 points, how can I find a quadratic equation that intersects all of these points?

Question

Given 3 points, $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, how might I find an equation intersecting all of these points? Given just 2 points, to find a linear equation, this is the formula: $$y\ =\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}x+\frac{\left(y_1+y_2-\left(x_1\left(\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}\right)+x_2\left(\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}\right)\right)\right)}{2}$$ In the same form of $y=...$, what is the formula for a quadratic equation with 3 points?

Answer 1

Note that by Lagrange's polynomial

$$L(x) = {y_1}\cdot{x - x_2 \over x_1 - x_2}\cdot{x - x_3 \over x_1 - x_3}+ {y_2}\cdot{x - x_1 \over x_2 - x_1}\cdot{x - x_3 \over x_2 - x_3}+ {y_3}\cdot{x - x_1 \over x_3 - x_1}\cdot{x - x_2 \over x_3 - x_2}$$

See some example of application here.

Answer 2

you can make the ansatz $$y=a^2+bx+c$$ and plug in all your points in this equation and then you must solve the system $$y_1=ax_1^2+bx_1+c$$ $$y_2=ax_2^2+bx_2+c$$ $$y_3=ax_3^2+bx_3+c$$ for $a,b,c$