Why coefficients have to be proportional for two quadratic functions to have the same roots?

Question

We have the next two quadratic functions:

$ ax^2 + bx + c = 0 $

$ mx^2 + nx + p = 0 $

If $ a/m = b/n = c/p $ then they have the same roots.

What is the intuition behind this statement?

Answer 1

Factor out the constant of proportionality from one equation, then divide each side of the equation by that constant, and you'll have the second equation.

ETA: To demonstrate, let $k = a/m = b/n = c/p$. Then we have \begin{align*} mx^2 + nx + p &= 0 \\ kax^2 + kbx + kc &= 0 \\ k (ax^2 + bx + c) &= 0 \\ ax^2 + bx + c &= 0 \end{align*}

Answer 2

Let $k$ be the constant of proportionality; i.e., let $k = \frac{a}{m} = \frac{b}{n} = \frac{c}{p}$. Then, $km = a$, $kn = b$, and $kp = c$; and

$$\begin{align*}ax^2 + bx + c = 0 & \iff (km)x^2 + (kn)x + kp = 0 \\ & \iff k(mx^2 + nx + p) = 0 \\ & \iff mx^2 + nx + p = 0.\end{align*}$$

Therefore, a root of the one quadratic is a root of the other.