Basic calculus problem - is this approach correct?

Hello,

I'm self teaching and working on the following question:

"The curve y = ax^2 + bx + c crosses the y-axis at the point (0,3) and
has a stationary point at (1,2). Find the values of a, b and c."

I can find that c = 3 easily. That's revealed by the point (0,3).

a and b are causing me difficulty though. My text has taught me that
the derivitive function will equal zero at a stationary point.
Therefore:

2ax + b = 0 at the stationary point. I know x = 1 at one of the
stationary points. Therefore 2a(1) + b = 0 and therefore b = -2a.

I want to know if the following step is valid or not: plugging b = -2a
into the original equation for when x = 1 to get a as follows:

2 = a(1)^2 - 2a(1) + 3 ---> a = 1.

and then finding b using:

2 = -1(1)^2 + b(1) + 3 ---> b = -2

To rephrase my question, does the relationship b = -2a found in the
derivative function also apply to the original function?

Regards,

Peter Barker
.

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