For a quadratic function f(x)=ax^2 + bx + c, the derivative is f'(x) = 2ax + b.

Varying c moves f up and down without affecting f’, so we want to find c that gives only one single solution.

They meet where f(x)=f'(x), or x^2 + (b-2a)x/a + (c-b)/a = 0, which has exactly one solution when c=a + b^2/(4a).

So the answer is: ax^2 + bx + a + b^2/4a

]]>