Completing the Square – Fast and Formal

Here is a strictly formal way to complete the square. We want to express ax^2+bx+c in the form a(x-h)^2+k.  We may want to do this for many reasons, for example, to graph a quadratic function (parabola), to solve a quadratic equation, to integrate a rational function or to solve a differential equation.

If we expand a(x-h)^2+k, we get ax^2-2ahx+ah^2+k. If ax^2+bx+c=ax^2-2ahx+ah^2+k for all x, and we know a,b \text{ and } c, we only need to solve two equations.  First solve b = -2ah for h. Note we already know a.  Then solve c = a^2h+k for k. Here we already know a\text{ and } h. Here are two examples.

Completing the Square Example I

Completing the Square Example II

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About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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