Completing the square
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Solve x^2 + 6x + 2 = 0 by completing the square.
Common mix-up
In this example, a student wrote x^2 + 6x + 9 = -2.
The student added 9 only to the left side, writing x^2 + 6x + 9 = -2 and unbalancing the equation.
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See why this works
What do you add to make x^2 + 6x a perfect square?
A perfect square like (x + 3)^2 is easy to solve. Completing the square builds one on purpose, right in the middle of an equation.
Add (b/2)^2 to both sides to create a perfect square trinomial.
Some quadratics are one small step away from a perfect square. In x^2 + 6x + 2 = 0, the x^2 + 6x part nearly matches (x + 3)^2 = x^2 + 6x + 9. The 9 is missing, and the first move is to isolate that pair: subtract 2 from both sides, leaving x^2 + 6x = -2.
Now make the square deliberately. Take half of 6, which is 3, and square it to get 9. Add 9 to both sides: x^2 + 6x + 9 = -2 + 9. Adding to one side only would break the equality, and that is the most common slip in this method.
The left side is now a perfect square, so write it as (x + 3)^2 = 7. Take the square root of both sides: x + 3 = +/- sqrt(7). The plus or minus appears because both a positive and a negative number square to 7. Finish by subtracting 3, giving x = -3 +/- sqrt(7).
Substitute to check. With x = -3 + sqrt(7), the square is 16 - 6*sqrt(7), and 6x is -18 + 6*sqrt(7). Adding 2: everything cancels and the result is 0. The exact answers keep the radical, which is cleaner than a decimal estimate and is what exact algebra asks for.
Keep this idea
Move the constant, add (b/2)^2 to both sides, factor into a perfect square, then take square roots and undo the leftover number.
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