###### Theorem 2.1 Quadratic Formula

Given the second-degree polynomial equation \(ax^2 + bx + c = 0\text{,}\) where \(a\neq0\text{,}\) solutions are given by

\begin{equation*}
x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}.
\end{equation*}

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In the previous section, we saw relatively simple WeBWorK questions. This section demonstrates how even very complicated WeBWorK problems can still behave well.

Here is a theorem that gives us a formula for the solutions of a second-degree polynomial equation. Note later how the WeBWorK problem references the theorem by its number. This seemingly minor detail demonstrates the degree to which WeBWorK and PreTeXt have been integrated.

Given the second-degree polynomial equation \(ax^2 + bx + c = 0\text{,}\) where \(a\neq0\text{,}\) solutions are given by

\begin{equation*}
x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}.
\end{equation*}

\begin{align*}
ax^2 + bx + c &= 0\\
ax^2 + bx &= -c\\
4ax^2 + 4bx &= -4c\\
4ax^2 + 4bx + b^2 &= b^2 - 4ac\\
(2ax + b)^2 &= b^2 - 4ac\\
2ax + b &=\pm\sqrt{b^2 - 4ac}\\
2ax &=-b\pm\sqrt{b^2 - 4ac}\\
x &=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}
\end{align*}