Skip to main content Contents Index Prev Up Next \(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}}
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Section A.2 The Quadratic Formula
Definition A.1 . The Quadratic Formula.
The quadratic formula gives the solution of the second-order polynomial equation
\begin{equation}
ax^2 + bx + c = 0\tag{A.2.1}
\end{equation}
i.e.
\begin{equation}
x=\frac{1}{2a}\left\{-b\pm\sqrt{b^2-4ac}\right\}\tag{A.2.2}
\end{equation}
The proof of the quadratic formula involves completing the square, see
Section A.1 . Starting from
\begin{equation}
ax^2 + bx + c = 0\tag{A.2.3}
\end{equation}
we immediately have
\begin{equation}
a \left(x+\frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c = 0\tag{A.2.4}
\end{equation}
so that
\begin{align}
\left(x+\frac{b}{2a}\right)^2
\amp = \frac{b^2}{4a^2} - \frac{c}{a}\tag{A.2.5}\\
\amp = \frac{b^2-4ac}{4a^2} .\tag{A.2.6}
\end{align}
Taking the square root of both sides and rearranging terms yields the quadratic formula in its standard form
(A.2.2) .
Definition A.2 . Degeneracy.
An
\(n\) th-order polynomial always has
\(n\) (complex) roots, but these roots may be repeated. In physics contexts, we call the physical quantity represented by the repeated root
degenerate , see the contexts of matrix eigenvalues
Section 4.2 and solutions of ODEs
Section 15.6 . A root that is repeated
\(m\) times is called
\(m\) -fold degenerate.
