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Visualizing Completing the Square


Alignments to Content Standards: A-REI.B.4.a

Task

Enrico has discovered a geometric technique for ''completing the square'' to find the solutions of quadratic equations. To solve the equation $x^2 + 6x + 4 = 0$, Enrico draws a square of dimension $x$ by $x$ and attaches 6 strips (3 of dimension 1 by $x$ and 3 of dimension $x$ by 1) to make the picture below:

Square_decomposition_23ff18eee25bd27fd0c49c7859855bfa

To complete the larger square, Enrico adds 9 squares of dimension 1 by 1. He has 4 of them in his initial expression so he needs five more as shown in the picture. So the picture represents the equation $$ (x+3)^2 = (x^2 + 6x + 4) + 5 $$

  1. Explain how Enrico's method helps find the roots of $x^2 + 6x + 4 = 0$.
  2. Help Enrico draw a picture for, and then solve, the equation $x^2 - 6x + 4 = 0$.

IM Commentary

The purpose of this task is to help provide geometric intuition for the algebraic process called ''completing the square.'' This is often done to help solve quadratic equations but the reason why the process is called completing the square is usually not highlighted. It turns out that when the algebraic expressions are modeled geometrically we find part of a square which is missing a small piece in order to be complete. For the two quadratic equations considered here, the roots are real. This process applies to all quadratic equations, however, and is one of the best known methods for deriving the quadratic formula. If students are comfortable with the examples given here, the teacher might want to explore what happens when the coefficients of the quadratic equation are variables (and hence work toward deriving the quadratic formula).

Solution

  1. To solve the equation $(x^2 + 6x + 4) = 0$ using $(x+3)^2 = (x^2 + 6x + 4) + 5$, Enrico can proceed as follows. The second equation can be rewritten as $$ x^2 + 6x + 4 = (x+3)^2 - 5. $$ So the solutions of $x^2 + 6x + 4 = 0$ are the same as the solutions to $(x+3)^2 - 5 = 0$. We can add 5 to both sides obtaining $$ (x+3)^2 = 5 $$ and then taking a square root on both sides gives $$ |x+3| = \sqrt{5}. $$ This has solutions $x = -3 + \sqrt{5}$ and $x = -3 - \sqrt{5}$.

  2. For the equation $x^2 - 6x + 4 = 0$ we can still start with an $x$ by $x$ square representing $x^2$. But this time we want to take away $3x$ from each side of the square. We can do this as in the picture below but we have taken away the 9 small squares (where the rectangles of area $x$ intersect) twice.

    Square_decomposition_e458398b21bdbd2178246452991005c1

    This means that the small square, of dimensions $x-3$ by $x-3$ is equal to the big square minus the 6 rectangles of dimension $x$ by $1$ plus 9 (to account for the second time these 9 squares are taken away). In equation form: $$ (x-3)^2 = x^2 - 6x + 9. $$ From here we find $$ (x-3)^2 - 5 = x^2 - 6x + 4. $$ Solving as we did in part (a) we find that $x = 3+\sqrt{5}$ and $x = 3-\sqrt{5}$ are the two roots of the quadratic equation $x^2 - 6x + 4 = 0$.

Howard Phillips says:

almost 3 years

I feel sorry for Enrico, as he seems to have been given a graphical method for his problem, and has "learned' it. To pass a test? What is shown is a graphical illustration of "completing the square", and follows from graphical illustrations of (x+2)(x+3) = x squared + 2x + 3x +6, and more fun, the same for (x-2)(x-3).
I hope that Enrico is familiar with all this, and also (x+a) squared = x squared + 2ax +a squared. Completing the square is quite a sophisticated idea, and to teach it as a "method" is not developing understanding in the pupil.

Cam says:

almost 3 years

I think you may be reading too much into the word "learned," or at least reading it too narrowly. I don't think the implication is that Enrico was taught this and only this as "the method" for completing the square -- perhaps he discovered this technique himself! Indeed, as I think you would agree, there are substantial benefits from understanding completing the square from multiple viewpoints. A graphical visualization might help some students internalize what's happening on the sometimes-oblique algebraic side.

Cam says:

almost 3 years

Actually, taking your comment to heart, I like the "discovered" interpretation much better, and have made the change accordingly. Thanks for the feedback!