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Section: A1.7.5

Deriving the quadratic formula

Use completing the square to derive the quadratic formula (A-SSE.B.3b$^\star$, A-REI.B.4a).

Now that students have practice with completing the square, they have all of the tools they need to look at solving quadratic equations more abstractly and derive the quadratic formula. For some students, the algebraic manipulation in this section will seem overwhelming, but for others this section can bring home the connectedness between all they have learned about solving quadratic equations thus far. The main purpose of this section is to derive and build a conceptual understanding of the quadratic formula before using it to solve equations in the next section.

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1 Building a General Quadratic Function

WHAT: In this scaffolded task students modify the general form of a quadratic expression and arrive at the quadratic formula. The task states that it is important for students to have seen the 'Building an explicit quadratic function' task prior to this one and it could be a nice review to start with that task here.

WHY: This activity serves as a synthesis not only of the tools gained in this unit, but of students’ algebra skills developed up to this point in their entire career. In working through this task, students should see the quadratic formula not as a “magic wand” to wave at quadratics that they are having trouble solving, but one that is grounded in completing the square and in familiar algebra. This task will help to build a conceptual understanding for the formula before the students begin to tackle procedural fluency with the formula in the next section (MP3).