Solve quadratic equations by completing the square
Understand and be able to use the method of completing the square to solve quadratic equations.
This section continues to develop the method of completing the square with a carefully sequenced set of activities. Initially students look at equations in which an expression of the form $x^2+2ax$ is clearly visible; then they consider equations in general form. By the end of the section students arrive at completing the square as a general method for solving quadratic equations, both executing the procedure and understanding how it works.
WHAT: This sequence of visual patterns leads to the function $f(x)= x^2+2x$ giving the number of squares as a function of the step number, $x$. Previously students solved equations of the form $f(x)=k$ for such sequences of patterns by “undoing” operations. Now they must use what they know to rewrite the equations first A-REI.A.1, A-REI.B.4a.
WHY: By returning to a difficult case of a familiar problem, this task reinforces the idea of completing the square as a culmination of progressively more sophisticated techniques. Visually, this pattern can also be used to show completing the square as the patterns can actually be manipulated into “squares” the same way that the equation is manipulated into a perfect square (but this is also addressed explicitly in sample activity 3.3) MP.4.
WHAT: This task provides a geometric visualization of completing the square. The visualization is a literal completing of the square: when the algebraic expressions are modeled geometrically we find part of a square which is missing a small piece in order to be complete A-REI.A.1, A-REI.B.4.
WHY: By connecting geometry with algebra this task helps build conceptual understanding for a topic that is often seen as procedural. The activity could be followed by some practice for procedural fluency with completing the square.
WHAT: This activity builds students’ ability to see an expression of the form $x^2+2ax$ and to recognize the constant that must be added to it to form a perfect square, through a carefully designed sequence of exercises MP.8. It then gives students practice transforming equations using this ability, with the steps for transformation getting successively more difficult. The sequence of equations leads students to understand completing the square and see that this method works for solving any quadratic equation A-REI.A.1, A-REI.B.4.
WHY: It gives students opportunities to make use of structure MP.7 and detect patterns MP.8 as they develop the method of completing the square. Note that the initial table comparing factoring with completing the square will only make sense for students familiar with the former method. It can be omitted without harm to the rest of the activity.