Conceptually(not algebraically), why do I get a quadratic equation when I subtract a linear equation form a quadratic equation.

Question

While simultaneously solving a quadratic equation and a linear equation, I noticed that their intercepts are actually equal to the x intercepts of another quadratic function. I find it hard to grasp that subtracting a linear equation from a quadratic equation results in a symmetric curve(parabola).

I understand that the result of the subtraction will give me a polynomial of degree 2 which forms a parabola, but why?

Answer 1

One way of looking at it is through finite differences. A quadratic function has the property that successive differences follow a linear progression. For example, with $f(n) = n^2 + n + 1$:

n  f(n)  difference
-------------------
0    1  
          2
1    3  
          4
2    7
          6
3   13
          8
   ...

A linear function has the property that successive differences are constant. Thus, adding a linear function to a quadratic function will shift the differences of the latter without disturbing their linearity.

The argument in the previous paragraph relies on an understanding that "constant + linear = linear", so in fact you can see that you can use the same reasoning inductively for adding polynomials of any degree.

Answer 2

A higher level understanding of this could come from the study of affine geometry. In affine geometry all conics are either ellipses, hyperbolas, or parabolas, and are not changed in character by any affine transformation. Your notion of

subtracting a linear equation from a quadratic equation

is a special case of an affine transformation and and hence transforms a parabola to another parabola.