In this lesson, students learn that completing the square can be used to solve any quadratic equation, including equations that involve rational numbers that are not integers. Students notice that the process of completing the square is the same when the equations involve messier numbers as when they have simple integers, but the calculations may be more time consuming and prone to error. An error-analysis activity highlights some common errors related to completing the square.
Although any equation can be solved by completing the square, equations that are really difficult to solve by this method are not included here. Students will solve such equations when they have access to the quadratic formula. What is important in this lesson is to recognize that putting a quadratic equation in the form of \((x+p)^2 =q\) allows them to solve it, but there are cases in which doing so may not always be the most efficient strategy.
Completing the square for quadratic expressions that are more elaborate encourages students to look for and make use of the same structure that helped them when they were working with less complicated expressions (MP7).
- Express any quadratic equation in the form $(x+p)^2=q$ and solve the equation by finding square roots.
- Generalize (orally) a process for completing the square to express any quadratic equation in the form $(x+p)^2=q$.
- Solve quadratic equations in which the squared term has a coefficient of 1 by completing the square.
- Let’s solve some harder quadratic equations.
- When given a quadratic equation in which the coefficient of the squared term is 1, I can solve it by completing the square.
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