Finding coefficients of quadratic formula given certain properties Given the quadratic equation $ax^2+bx+c$  , how do you find $a,b$ and $c$ given you know: 
the gradient of the curve at the $y$ intercept
the equation of the tangent at point $P$
the gradient of the normal at point $P$
I haven’t included the specific equations and stuff as I would like to work it out myself, I just need to know what steps to take. 
 A: Hint. In other words, you want to determine the curve $y=ax^2+bx+c,$ given that
(1) $y'=m$ at $x=0,$
(2) $y'=n$ at $x=p,$
(3) $y=q$ when $x=p.$
You have three linear equations in $a,b$ and $c.$
PS. The value $n$ is equal to $-1/n,$ the gradient of the normal at $P(p,q).$
Can you continue now?
A: Let $f(x) = ax^2 + bx + c$ be the curve $C$ in question.
If we know the gradient of $C$ at the $y$-intercept (i.e. where $x=0$) is $m_0$, then that is the same as saying we know that $f'(0) = m_0$.
If we know the equation of the line $L$ which is tangent to $C$ at the point $P$ with co-ordinates $(x_P, y_P)$, then we know two things:


*

*$P$ lies on $C$, and thus we know that $f(x_P) = y_P$; and

*the equation describing $L$ can be written in the form $y = m_P x + d$, whence $m_P$ is the gradient of this line. Thus, we know that $f'(x_P) = m_P$.


Knowing the gradient of the normal to $C$ at $P$ provides no extra information, since this is guaranteed to be equal to $\frac{-1}{m_P}$.
You can substitute $f(x)$ and $f'(x)$ into the three "we know that" statements given above, and use the resulting system of three linear equations to solve for $a$, $b$, and $c$.
