What is the meaning of a discriminant graphically?

Question

I know that if $b^2-4ac>0$, then there are real solutions and so on.
But for any quadratic function, where is the discriminant present in the plot?
For example, $x^2-8x+7=f(x)$.

Now how is $\Delta=36$ associated with the graph?

Answer 1

The square root of the discriminant is the distance between the zeros in case we have $a=1$. In your plot we have $$7-1 = 6=\sqrt{36}=\sqrt{\Delta}.$$

Proof:

We have $$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$$ so $$x_1-x_2= \frac{-b+\sqrt{\Delta}}{2a} - \frac{-b-\sqrt{\Delta}}{2a} = \frac{2\sqrt{\Delta}}{2a}= \frac{\sqrt{\Delta}}{a}=\sqrt{\Delta}.$$

Answer 2

If you think about the quadratic formula:

$$ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$

then the difference between the roots is:

$$\frac{\sqrt{b^2-4ac}}{a} $$

i.e. $\sqrt{\Delta}/a$.