Completing the Square Versus The Quadratic Formula

My college algebra class is learning to find the zeros of polynomial functions, so we are solving lots of quadratic equations.  If the quadratic expression is not easily factorable, I have been saying. “Just use the Quadratic Formula.”  Here is an example.

Quadratic Formula with Even Linear Coefficient

Quadratic Formula with Even Linear Coefficient

Note all of the awkward simplifying at the end.  It dawned on me that if the coefficient of the x term is even, completing the square is much easier as can be seen here.

Completing the Square  with Even Linear Coefficient

Completing the Square with Even Linear Coefficient

The quadratic formula is simply completing the square as a compact expression, but when the coefficient of the x term is even, completing the square just seems cleaner.  Not so with odd coefficients as can be seen here.

Completing the Square  with Odd Linear Coefficient

Completing the Square with Odd Linear Coefficient

Versus:

Quadratic Formula with Odd Linear Coefficient

Quadratic Formula with Odd Linear Coefficient

So, at least in my classroom demonstrations:  Even linear coefficients – complete the square; Odd linear coefficients – quadratic formula.

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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.
This entry was posted in Curriculum, Pedagogy, Teaching. Bookmark the permalink.

One Response to Completing the Square Versus The Quadratic Formula

  1. Anonymous says:

    Thank you for this posting and clear examples. I automatically go to the quadratic formula, but now have students who only approach a problem by completing the square. As long as they can find the roots, I am fine with either method. I was just curious about any discussions about the differences in the methods.

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