Proof of Quadratic Theorem

Written below is the proof of the quadratic theorem:

Quadratics are of the form

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

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:

$ax^2+bx+c=0$

$a(x^2+\frac{b}{a}x)+c=0$

At this point, we use a method called completing the square.

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

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

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

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

\[x+\frac{b}{2a}=\frac{\pm\sqrt{b^2-4ac}}{2a}\]

\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

The proof of this theorem is much more important than the theorem itself. Completing the square, as used above, is an incredibly useful tool and shows up in many competitions. Mastering the use of the tool is an extremely practical skill, and must be learned.

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