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#+begin_center
*Finding Roots of a Quadratic Equation*
#+end_center

The process of /"completing the square"/ makes use of the algebraic
identity

    #+NAME: step0
    $$x^2 + 2hx + h^2 = (x+h)^2$$

which represents a well-defined algorithm that can be used to solve
any quadratic equation.

Starting with a quadratic equation in standard form, $ax^2 + bx + c = 0$

1. Divide each side by $a$, the coefficient of the squared term.

       #+NAME: step1
       $$x^2+{ b over a }x+{ c over a } = 0$$

2. Subtract the constant term $c slash a$ from both sides.

       #+NAME: step2
       $$x^2+{ b over a }x = -{ c over a }$$

3. Add the square of one-half of $b slash a$, the coefficient of $x$, to both sides. This /"completes the square"/, converting the left side into a perfect square, as in [[step0]].

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

       #+NAME: step3
       $$x^2 + 2({ b over { 2a } })x + ({ b over { 2a } })^2 = -{ c over a } + ({ b over { 2a } })^2$$

4. Write the left side as a square and simplify the right side if necessary.

       #+NAME: step4
       \begin{equation}
       matrix {
         (x + b over { 2a })^2 # {} = {} # - { c over a } + { b^2 over { 4a^2 } }
         ##
         ~ #                     {} = {} # (-4ac + b^2) over { 4a^2 }
       }
       \end{equation}

5. Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side.

       #+NAME: step5
       $$matrix{ (x + b over { 2a }) # {} = {}# +- sqrt{ { -4ac + b^2 } over { 4a^2 } } ## {} # {} = {} # +- { sqrt{ -4ac + b^2 } over { 2a } } }$$

6. Solve each of the two linear equations.

       #+NAME: step6
       $$matrix{ x # {} = {} # -b over { 2a } +- { sqrt{ -4ac + b^2 } over { 2a } } ## {} # {} = {} # { -b +- { sqrt{ -4ac + b^2 }  } } over { 2a } }$$