The $x$-coordinate of the center of a parabola $ax^2 + bx + c$ is $$-\frac{b}{2a}$$
If we look at the quadratic formula
$$\frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$
we can see that it specifies two points at a certain offset from the center
$$-\frac{b}{2a} \pm \frac{\sqrt{b^2 – 4ac}}{2a}$$
This means that $\frac{\sqrt{b^2 – 4ac}}{2a}$ is the (horizontal) distance from the vertex to the roots. If I squint, the two squared-ish quantities being subtracted under a square root sign reminds me of the Euclidean distance formula
$$\sqrt{(x_0 – x_1)^2 + (y_0 – y_1)^2}$$
Is there a connection? If not, is there any intuitive or geometric reason why $\frac{\sqrt{b^2 – 4ac}}{2a}$ should be the horizontal distance from the vertex to the roots?
Best Answer
Suppose that $a\ne 0$ and $b^2-4ac\geq 0$. Let $f(x)=ax^2+bx+c$ and $$ x_0= -\frac{b}{2a},\; x_1=\frac{-b+\sqrt{b^2-4ac}}{2a}, \;x_2=\frac{-b-\sqrt{b^2-4ac}}{2a}. $$ It follows that $$ y_0=f(x_0)=\frac{-b^2+4ac}{4a},\;y_1=f(x_1)=0, \; y_2=f(x_2)=0. $$ The distances from vertex to roots: $$ d_1=\sqrt{(x_1-x_0)^2+(y_1-y_0)^2}=\sqrt{\left(-\frac{b}{2a}-\frac{-b+\sqrt{b^2-4ac}}{2a}\right)^2+\left(\frac{-b^2+4ac}{4a}\right)^2}=\frac{\sqrt{(b^2-4ac)^2+4(b^2-4ac)}}{4|a|}>\frac{\sqrt{b^2-4ac}}{2|a|}, $$ $$ d_2=\sqrt{(x_2-x_0)^2+(y_2-y_0)^2}=\sqrt{\left(-\frac{b}{2a}-\frac{-b-\sqrt{b^2-4ac}}{2a}\right)^2+\left(\frac{-b^2+4ac}{4a}\right)^2}=\frac{\sqrt{(b^2-4ac)^2+4(b^2-4ac)}}{4|a|}>\frac{\sqrt{b^2-4ac}}{2|a|}. $$