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https://en.wikipedia.org/wiki/Quadratic_formula#Derivations_of_the_formula
Eq. \ref{eq:8882098} is an initial equation. \begin{equation} ax^2 + bx + c = 0 \label{eq:8882098} \end{equation} Divide both sides of Eq. \ref{eq:8882098} by \(a\); yields Eq. \ref{eq:7777621}. \begin{equation} x^2 + (b/a)x+(c/a) = 0 \label{eq:7777621} \end{equation} Subtract \(c/a\) from both sides of Eq. \ref{eq:7777621}; yields Eq. \ref{eq:1212129}. \begin{equation} x^2 + (b/a)x = -c/a \label{eq:1212129} \end{equation} Add \((b/(2 a))^2\) to both sides of Eq. \ref{eq:1212129}; yields Eq. \ref{eq:1120000}. \begin{equation} x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2 \label{eq:1120000} \end{equation} Change variable \(h\) to \(b/(2 a)\) in Eq. \ref{eq:9091270}; yields Eq. \ref{eq:1239010}. \begin{equation} x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2 \label{eq:1239010} \end{equation} Simplify Eq. \ref{eq:1239010}; yields Eq. \ref{eq:1734000}. \begin{equation} x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2 \label{eq:1734000} \end{equation} Eq. \ref{eq:9091270} is an initial equation. \begin{equation} x^2 + 2 x h + h^2 = (x + h)^2 \label{eq:9091270} \end{equation} LHS of Eq. \ref{eq:1120000} is equal to LHS of Eq. \ref{eq:1734000}; yields Eq. \ref{eq:2985412}. \begin{equation} (x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2 \label{eq:2985412} \end{equation} Take the square root of both sides of Eq. \ref{eq:2985412}; yields Eq. \ref{eq:6608123} and Eq. \ref{eq:6608102}. \begin{equation} x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)} \label{eq:6608123} \end{equation} \begin{equation} x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)} \label{eq:6608102} \end{equation} Subtract \(b/(2 a)\) from both sides of Eq. \ref{eq:6608123}; yields Eq. \ref{eq:4433112}. \begin{equation} x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a)) \label{eq:4433112} \end{equation} Subtract \(b/(2 a)\) from both sides of Eq. \ref{eq:6608102}; yields Eq. \ref{eq:2657355}. \begin{equation} x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a)) \label{eq:2657355} \end{equation} Simplify Eq. \ref{eq:2657355}; yields Eq. \ref{eq:8811221}. \begin{equation} x = \frac{-b-\sqrt{b^2-4ac}}{2 a} \label{eq:8811221} \end{equation} Simplify Eq. \ref{eq:4433112}; yields Eq. \ref{eq:8761200}. \begin{equation} x = \frac{-b+\sqrt{b^2-4ac}}{2 a} \label{eq:8761200} \end{equation} Eq. \ref{eq:8811221} is one of the final equations. Eq. \ref{eq:8761200} is one of the final equations.