Areas of Math were developed to deal with problems which couldn't be tackled with preexisting methods. So, for example, 2D geometry was used to solve measuring problems with land, basic algebra for figuring out basic physical quantities, etc.
What physical or intuitive problem is solved by quadratic equations and their graphs?
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The Babylonians solved problems, in many cases stated geometrically, essentially using quadratic equations. For example, the area of a square diminished by the side of the square is (a given number); find the side of the square. I don't think it's at all clear what were the practical problems (if any) that motivated the discovery of these methods: what we have (on numerous clay tablets) are essentially school exercises for teaching what was already well-developed material.
Throw an object from height $h_0$ with upward velocity $v_0$. Its height at time $t$ is given by the quadratic $$h(t)=-\frac{g}{2}t^2+v_0t+h_0$$ where $g$ is the gravitational constant.