Finding points of intersection of two quadratics: How do you know what '$x$ value' to substitute into what 'quadratic equation'

Question

I am trying to find the points of intersection of $y = x^2 -3x$ and $y = 3x^2 - 5x -24$

I first set them equal to each other:

$$2x^2 -2x -24 = 0$$

Then I solved it:

$$\begin{align}x &= 4 \\ x &= -3\end{align}$$

Now as I currently understand you must substitute those two $x$ values into their respective quadratic equations in order to find the $y$ co-ordinate for the points of intersections. E.g

$$\begin{align}fx(4) &= 3x^2 - 5x -24 \\ fx(-3) &= x^2 -3x\end{align}$$

How do I know what $x$ value to substitute into what equation?

Answer 1

It doesn't matter which one you substitute it in since it is a point of intersection, thus the $y$-coordinate of both graphs will be the same.

For example, let's try finding the coordinates of intersection where $x=4$:

$$f(x)=x^2-3x \Rightarrow f(4)=16-12=4$$ Now, let's try this with the other function: $$f(x)=3x^2-5x-24 \Rightarrow f(4)=48-20-24=4$$ Which is the same answer: $(4,4)$.

Here is a graph showing the intersection:

Answer 2

Since these are the points where $x^2 - 3x = 3x^2 - 5x - 24$, you can substitute these $x$ values into either expression and you should get the same $y$ value: $y=18$ for $x=-3$ and $y=4$ for $x=4$.

Answer 3

After once you equated a common $y$ you can find $y,$ for any one parabola you may choose. Sketched is a third possible parabola through fixed points $(-3,18),(4,4) $