Find equation of parabola given two points and known area under curve

Question

I have a parabola that passes through two points: $(0,50)$ and $(10, 10)$. I also know that the area under the curve, between these two points, is $300$.

Is there a method by which I might find equations that satisfy these requirements?

Thanks much!

---

P.S. Would it be accurate to rewrite these givens in the form: $f(0) = 50$, $f(10) = 10$, and $\int_{0}^{10} f(x)\, dx = 300$ ?

Answer 1

$$f\left( x \right) =a{ x }^{ 2 }+bx+c\\ f\left( 0 \right) =50\Rightarrow c=50\\ f\left( 10 \right) =100a+10b+50=10\quad \Rightarrow \quad 10a+b=-4$$ Now solve system equations to find $a$ and $b$ $$\int _{ 0 }^{ 10 }{ \left( a{ x }^{ 2 }+bx+50 \right) dx=300 } \\ 10a+b=-4$$