The normal to the curve y = a x²+ b x at the point where x = 1 has gradient 1
and passes through the point (4,5). Find a and b.
the tangent has slope 2ax+b
so at x=1, the normal has slope -1/(2a+b) = 1
the equation is thus
y = x+1
So we now know that
2a+b = -1
a+b = 2
so a = -3 and b=5
y = -3x^2 + 5x