Hello, and welcome to MHB! (Wave)
Let's first look at a graph of the two given functions and the given points:
View attachment 8479
From the graph, it appears the given points are where the two functions \(f\) and \(g\) intersect. Let's look at a table:
[table="width: 300, class: grid"]
[tr]
[td]\(x\)[/td]
[td]\(f(x)\)[/td]
[td]\(g(x)\)[/td]
[/tr]
[tr]
[td]-2[/td]
[td]24[/td]
[td]16[/td]
[/tr]
[tr]
[td]
-1[/td]
[td]
10[/td]
[td]
10[/td]
[/tr]
[tr]
[td]0[/td]
[td]0[/td]
[td]6[/td]
[/tr]
[tr]
[td]1[/td]
[td]-6[/td]
[td]4[/td]
[/tr]
[tr]
[td]2[/td]
[td]-8[/td]
[td]4[/td]
[/tr]
[tr]
[td]3[/td]
[td]-6[/td]
[td]6[/td]
[/tr]
[tr]
[td]4[/td]
[td]0[/td]
[td]10[/td]
[/tr]
[tr]
[td]5[/td]
[td]10[/td]
[td]16[/td]
[/tr]
[tr]
[td]
6[/td]
[td]
24[/td]
[td]
24[/td]
[/tr]
[tr]
[td]7[/td]
[td]42[/td]
[td]34[/td]
[/tr]
[/table]
The table would indicate that the two given points are where the two functions intersect. Now, we can verify this algebraically as follows:
$$f(x)=g(x)$$
$$2x^2-8x=x^2-3x+6$$
Collect everything on the LHS:
$$x^2-5x-6=0$$
Factor:
$$(x+1)(x-6)=0$$
And so the values of \(x\) for which the two functions are equal are:
$$x\in\{-1,6\}$$
Let's verify by finding the values of the functions for those two values of \(x\):
$$f(-1)=2(-1)^2-8(-1)=2+8=10$$
$$g(-1)=(-1)^2-3(-1)+6=1+3+6=10$$
$$f(6)=2(6)^2-8(6)=72-48=24$$
$$g(6)=(6)^2-3(6)+6=36-18+6=24$$
And so we may conclude that the functions:
$$f(x)=2x^2-8x$$ and $$g(x)=x^2-3x+6$$
intersect at the points:
$$(-1,10)$$ and $$(6,24)$$
Does all that make sense?
