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Written below is the proof of the quadratic theorem:

$f(x)=ax^2+bx+c=0$

The roots or zeros of the quadratic are the values in which we can input as the $x$ value in the function to output $0$. The possible $x$ values that go into a function is referred to as the domain, and the possible $y$ values that come out of a function are referred to as the range. Be cautious to not confuse this word with the term range in a set of data, or the absolute difference between the greatest and least terms of a set.

Here is the proof of how to find the roots of a quadratic:

$ax^2+bx+c=0 a(x^2+\frac{b}{a}x)+c=0$ (Error compiling LaTeX. ! Missing $inserted.) At this point, we use a method called completing the square.$a(x^2+\frac{b}{a}x)=-c

a(x+\brac{b}{2a})^2=-c+\frac{b}{4a}

(x+\frac{b}{2a})^2=\frac{-c}{a}+\frac{b}{4a^2}

(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}

x+\frac{b}{2a}=\frac{\pm(b^2-4ac)}{2a}

x=\frac{-b\pm(b^2-4ac)}{2a}$(Error compiling LaTeX. ! Missing$ inserted.)