### Instruction: Two parabolas are the graphs of the equations $y=2x^2-7x+1$ and $y=8x^2+5x+1$. Give all points where they intersect.  List the points in order of increasing $x$-coordinate, separated by semicolons.### Response: To find the points of intersection, I need to solve the system of equations $y=2x^2-7x+1$ and $y=8x^2+5x+1$.
I can do this by setting the expressions for $y$ equal to each other and simplifying: $2x^2-7x+1=8x^2+5x+1$.
Subtracting $2x^2-7x+1$ from both sides, I get $0=6x^2+12x$.
Dividing both sides by $6$, I get $0=x^2+2x$.
Factoring out $x$, I get $0=x(x+2)$.
Using the zero product property, I get $x=0$ or $x=-2$.
These are the $x$-coordinates of the points of intersection, but I also need the $y$-coordinates.
To find the $y$-coordinate when $x=0$, I can plug in $x=0$ into either equation for $y$.
I'll use the first one: $y=2(0)^2-7(0)+1=1$.
So one point of intersection is $(0,1)$.
To find the $y$-coordinate when $x=-2$, I can plug in $x=-2$ into either equation for $y$.
I'll use the second one: $y=8(-2)^2+5(-2)+1=33-10+1=24$.
So another point of intersection is $(-2,24)$.
These are the only two points of intersection, since a parabola can only intersect another parabola at most twice.
To list them in order of increasing $x$-coordinate, I start with the one with the smaller $x$-value.
That is $(-2,24)$.
Then I put a semicolon and write the next one.
That is $(0,1)$.