The first portion of your work is correct. The graphs of the two equations intersect at x=-3 and x=2. However, there is a discrepancy with the y-coordinates of the intersections. Let's go over it to clarify.
The given equations are:
`y=x^2`
`y= 6-x`
To solve for their intersections, set the two y's equal to each other.
`y = y`
`x^2=6-x`
`x^2+x-6=0`
`(x+3)(x-2)=0`
`x+3=0` `x-2=0`
`x=-3` `x=2`
Then, plug-in the x values to either of the original equations. Let's use the second equation y=6-x.
So, when x=-3, the y value is:
`y=6-(-3)=9`
And, when x=2, the y value is:
`y= 6-2=4`
Hence, the points of intersection are (-3,9) and (2,4).
Next, to get the distance between these two points, apply the formula:
`d = sqrt((x_2-x_1)^2+(y_2-y_1)^2)`
`d=sqrt((2-(-3))^2+(4-9)^2)`
`d=sqrt(5^2+(-5)^2)`
`d=sqrt50`
`d=5sqrt2`
Therefore, the distance between the points of intersection of the graphs of the two given equations is `5sqrt2` units.