$$\definecolor{quadratic}{RGB}{ 11, 153, 11 } \definecolor{fraction}{RGB}{ 0, 0, 255 } \definecolor{constant}{RGB}{ 180, 180, 0 } \definecolor{vertasymptote}{RGB}{ 255, 0, 0 }$$

Suppose $a, b, c \in \mathbb{R}$. When $a \ne 0$, the equation $ax^2 + bx + c = 0$ hast two complex solutions (provided they are counted with their multiplicities). The solutions can be found using the quadratic formula: $x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$ The expression under the radical, $D = b^2 - 4 ac$, is called the discriminant. When the discriminant is positive, there are two real roots of multiplicity one (like in the example pictured below). If the discriminant is zero, there is one real root of multiplicity two. If the discriminant is negative, there are two complex conjugate roots of multiplicity one.

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Graph of the function $f( x ) = {\color{quadratic}x^2 - 5x + 3}$.

Rational functions have vertical asymptotes at the values of $x$ where the denominator polynomial has roots, and horizontal asymptotes given by the constant which is the result of the long division of the numerator by the denominator.

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Graph of the function $f( x ) = {\color{constant}1} + {\color{fraction}\frac{ 2 }{ 5x - 10 }}$. It has the vertical asymptote ${\color{vertasymptote} x = 2 }$ and horizontal asymptote ${\color{constant} y = 1 }$.

If the result of long division is not a constant but a polynomial, that polynomial's graph becomes the horizontal asymptote (of sorts) for the rational function in question.

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Graph of the function $f( x ) = {\color{quadratic}x^2 - 5x + 3} + {\color{fraction}\frac{ 2 }{ 5x - 10 }}$. It has the vertical asymptote ${\color{vertasymptote} x = 2 }$ and horizontal (in a way... it is not really a line) asymptote ${\color{quadratic} y = x^2 - 5x + 3}$.