# Why bother completing the square to find the minimum/maximum of a quadratic function?

Given a question like

Find the coordinates of the minimum point on the curve $y=3x^2+2x+9$.

students are often taught to solve this by completing the square.

The class I am currently teaching this to find completing the square difficult, and I think they would find it much easier to use the quadratic formula to find the roots and then average them to find the $x$ coordinate of the minimum point.

Why is finding minima/maxima of quadratic functions not usually taught like this? What are the advantages of solving these problems by completing the square?

• Why even find both roots? The formula for the $x$-location of the vertex is what you get when you remove the radical part of the quadratic equation. (Also, do they still find it easier when the roots are complex?)
Jun 27 '18 at 18:07
• What are the advantages of solving these problems by completing the square? --- Much needed practice with algebraic manipulation skills? Jun 27 '18 at 18:44
• I guess for problems where the roots are complex, it will be a little trickier, no? Plus what Dave said. Jun 28 '18 at 17:16

it much easier to use the quadratic formula to find the roots and then average them to find the x coordinate of the minimum point
1. It doesn't seem easier at all.
2. How did you derive the quadratic formula or is it black magic
3. Why does the image of the average of the roots give the extremum?
4. Completing the square are usually presented way before the quadratic formula (depends on the country)
5. It'll help later when studying circles and conic sections in analytic geometry.

• So basically students don't know where the quadratic formula came from and why should they average the roots. They just have to obey the commandments! Jun 27 '18 at 17:40
• Though you say your students won't study conic sections, most will study functions. Completing the square and getting the quadratic into standard form will help them understand shifting and stretching functions later (i.e., knowing what $y = a f(b(x-h)) + k$ only by knowing what $y=f(x)$ looks like).
– ncr
Jun 27 '18 at 20:42
• I especially agree with your point (3) - it is easy to explain why completing the square gives the extremum, but is not clear why averaging the roots does. Also, another application of completing the square: evaluate $\int\frac{1}{x^2+2x+2}dx$. Jun 28 '18 at 11:20

I don't think that I ever even heard of completing the square until graduate school when I was forced to teach it along with a number of other semi-archaic things like synthetic division.

I think that you can make a case for completing the square as a technique for deriving the quadratic formula and possibly for answering the question "why do all quadratics have parabola graphs". Now, once you actually have those, there may not be much of a case for having them do it by scratch again and again. The quadratic formula really is quite effective; why not use the tools we are given?

The question of "why parabolas" is really given by one of the formulas you get along the way: $$ax^2+bx+c = a\left(x+\frac{b}{2a}\right)^2 - \frac{b^2-4ac}{4a}$$ and observing how it tells you that the graph must be a shift and scale of the standard $x^2$ graph. That's worthwhile and you can push the idea along to cubics! There, you have the notion of writing a general cubic in the form of a depressed cubic by a similar stretch and shift. The new, depressed cubic now looks like $$t^3 +pt+q.$$ Notice the lack of a quadratic term. You can now figure out what any kind of cubic graph looks like by playing around with $p$ and $q$.

• Have they stopped teaching completing the square? Back in the dark ages, we were taught it in algebra I in eighth grade (lower secondary). Personally, I can never remember the quadratic formula so, decades later, I still find it easier to complete the square. Actually, completing the square on $ax^2+bx+c=0$ is how I recover the quadratic formula! Jun 27 '18 at 21:53
• There is nothing "archaic" about completing the square. For example, it will be needed by engineering students in upper level classes when they compute inverse Laplace transforms to analyze electric circuits. For another, the algebraic proof of the Cauchy-Schwarz inequality is a variant of the idea of completing the square. More generally, many basic analytic inequalities are obtained by the same idea - manipulate part of an algebraic expression so that it appears as a square, and hence is necessarily nonnegative. Jun 28 '18 at 7:55
• That it shows up, incidentally, years removed from when it was taught doesn't seem like a great argument for spending time on the technique in high school or in college algebra.
• @Adam: The Laplace transform and Cauchy-Schwarz inequality are hardly "incidental". In any case, completing the square can be understood as an application of translation of graphs of functions - what it does is represent any quadratic function as a translate (horiztonal and vertical) of the graph of $\pm x^{2}$ - and this is an idea with a lot of content directly relevant at the level at which it is taught. Jun 28 '18 at 20:40