Linear algebra with with unknown coefficientt for parabola

[Basheesh]

Homework Statement

Find the numbers a0, a1 and a2 so that the parabola with the equation:

y = a0 + a1 x + a2 x²

Goes through the points:
(2, −3), (9, 4), and (t, 4)

For any value of t, specify the amount of solutions

Homework Equations

y = a0 + a1 x + a2 x²

The Attempt at a Solution

I tried setting this up as a system of linear equations:

a0 + 2a1 + 4a2 = -3
a0 + 9a1 + 81a2 = 4
a0 + ta1 + t²a2 = 4

Solving for the 2 uppermost equations I get:

a0 = 18a2 - 5
a1 = -11a2 + 1

plugging these into the 3rd equation, I get:

18a2 -5 + t(11a2 +1) + a2t²

Which I can simplify to:

t = (9 - 18a2) / (11a2 + 1 0 a2t)

But I am then unsure of where to go from there, or whether this is even the correct approach to take.

Ps. Sorry if this is a little messy, I am new to the forum and unsure of how to correctly format the math.

 

[vela]

That's a good start. Instead of trying to solve for t, solve for a₂ in terms of t. Then you can find a₀ and a₁ in terms of t.