The problem asks us to find the intersection points of two parabolas, given by the equations $y = 5x - x^2$ and $y = x^2 - x + 4$. We need to solve for $x$ and $y$ where the two equations are equal.

1. Problem Description

The problem asks us to find the intersection points of two parabolas, given by the equations $y = 5x - x^2$ and $y = x^2 - x + 4$. We need to solve for $x$ and $y$ where the two equations are equal.

2. Solution Steps

First, we set the two equations equal to each other:

$5x - x^2 = x^2 - x + 4$

Next, we rearrange the equation to get a quadratic equation equal to zero:

$0 = x^2 - x + 4 - 5x + x^2$

$0 = 2x^2 - 6x + 4$

We can simplify the equation by dividing by 2:

$x^2 - 3x + 2 = 0$

Now we factor the quadratic equation:

$(x - 1)(x - 2) = 0$

This gives us two possible values for $x$:

$x = 1$ or $x = 2$

Now we substitute these values of $x$ into either of the original equations to find the corresponding $y$ values. Let's use $y = 5x - x^2$:

If $x = 1$, then $y = 5(1) - (1)^2 = 5 - 1 = 4$.

If $x = 2$, then $y = 5(2) - (2)^2 = 10 - 4 = 6$.

So the intersection points are $(1, 4)$ and $(2, 6)$.

3. Final Answer

The intersection points are $(1, 4)$ and $(2, 6)$.