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Completing the square means we want one side of the equation to match left side of the pattern: . Matching this pattern is easier if the leading coefficient is 1 so I'll divide both sides by a:
Now comes the tricky part. We need to figure what third term we need on the left side to match the pattern. With a leading coefficient of 1, this is a little easier. The third term we need is the square of 1/2 of the coefficient of x. The coefficient of x is b/a and 1/2 of this is b/2a and the square of b/2a is . This is the third term we need. So we will create it by adding it to both sides of the equation:
We now have the left side matching the pattern. So we can rewrite it as a perfect square:
(If you don't see this, multiply out the left side and see if you get ). We have now completed the square. The rest is solving for x.
On the right side we need to do some simplifying:
Get the denominators equal so we can add:
Add:
or
Now we can find the square root of each side:
On the right side we can simplify:
(We use instead of because we will end up with both the positive and negative values of the right side anyway.) Now we can simplify the left side:
Solving absolute value equations requires two equations: or
Now we add -b/2a to each each side: or
The two fractions on the right side of both equations conventiently have the same denominator so we can add them: or
The shorthand for these two equations is:
which is the familiar quadratic formula!
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