Lately I have been trying to think about equations in terms of graphs a lot and I stumbled on equation for solutions of quadratic equation, and I could not understand everything about it.
I can understand that $ \frac{-b}{2a}$ is supposed to result in the lowest point of the graph, but I can't quite wrap the head around how $ \sqrt{({\frac{b}{2a}})^2 – \frac{c}{a}} $ results in the distances of the zeroes from that lowest point. I had like some intuition behind what goes in there. Thanks!
Best Answer
Look at the following visual:
Let, $OA=\alpha$ and $OB=\beta$
By power of point for a circle, the length of the red line $=\sqrt{OA\cdot OB}=\sqrt{\alpha\cdot \beta}=\sqrt\frac ca$
Also, by Pythagoras Theorem, the length of the red line is $\sqrt{OC^2-AC^2}=\sqrt{(\frac{\alpha +\beta}{2})^2-AC^2}=\sqrt{(\frac{-b}{2a})^2-AC^2}$
Equate both the expressions to get your desired result.