yes, it is possible

example:

no, it is not possible.b. Three intercepts

note that the x-intercepts of a polynomial are the roots of the polynomial. and in general, an degree polynomial can have AT MOST n roots.

specific to the case of parabolas:

a parabola is a polynomial of the form

the x-intercepts are given by the roots of the equation, namely, the solutions to

it can be shown that the roots are given by . This is known as the quadratic formula, it can be derived by completing the square.

clearly there are three cases:

case 1: if we have TWO distinct, real roots.

case 2: if we have ONE real root (with a multiplicity of 2).

case 3: if , we have TWO complex roots

Thus we see, a quadratic can have at most two roots, which are the x-intercepts

if your professor demands a more concrete proof, you can look up the derivation of the quadratic formula here. it uses completing the square, which you will learn how to do soon, if you haven't already

It's 2am and i'm tired, so maybe there's an easier way to do this than what i'm seeing. but i'd set up a system of three equations.

The other part of the problem:

The vertex of a parabola is (5,3) and the parabola passes through the point (8,10). Find a third point that lies on the parabola?

I know that the vertex is the lowest point. Is there an equation that I would use to find the third point?

indeed, there is a formula for the vertex of a parabola (which by the way can be thehighestORlowestpoint of the parabola). the formula gives the x-coordinate for the vertex.

the x-coordinate for the vertex of a parabola of the form is given by:

we will use this formula to solve this problem.

Here's the information we have:

(5,3) ---vertex

(8,10) ----a point the parabola passes through

Let the parabola be of the form

since we have the vertex is (5,3), we know two things:

--------the vertex formula.

...........................(1)

also, when x = 5, y = 3. so we can plug this into our general form to get:

................(2)

since we have (8,10), we know, when x = 8, y = 10. this means

..................(3)

So now we have the system of equations:

...........................(1)

................(2)

..............(3)

From equation (1), we see that b = -10a, substitute this for b in equations (2) and (3) to get two new equations.

........................(4)

..........................(5)

.....................(4) - (5)

but,

also, recall that

Thus, our quadratic is:

Wow! That's weird quadratic. Hopefully there are more than one solutions and someone else can find a nicer one