Help finding the equation of this parabola.

Hint: There is another place you can find where y=18, the parabola is symmetrical about the line x=2

...

You've got to understand the mathematical information you're using to try to get the values of a,b,c.

You've correctly used the fact that the intercept is 18 to get c=18, which is good.
But then you used the fact that the point (2,10) lies on the parabola. Notice how just this information gives rise to many different parabolas, all with intercept 18 and passing through (2,10).

The key piece of information in the diagram that makes the parabola unique (and hence you can find the exact values of a,b,c) is that (2, 10) is the turning point of the parabola; it's not just a point that lies on the parabola.

So use what you know about turning points to achieve this: "I think I'm meant to come across a similar phrase that I can use simultaneous equations to find a and b."

As you can see there is another point you can derive, (4,18), you can use this to solve a pair of simultaneous equations with x and y

This was one way, thank you.

Write it in the form y=ax^2 + bx + c



Substitute in x=0 and you get ... c = 18

Substitue x and y in from the turning point and I got
2a + b = -4

I think I'm meant to come across a similar phrase that I can use simultaneous equations to find a and b. But as there are no roots, I'm not sure what to do. Any guidance please