Solve radical equations
• Solve radical equations and equations with fractional exponents (A-REI.A.2).
• Note extraneous solutions and describe where they came from(A-REI.A.2).
In this section, students will move from reasoning about expressions with rational exponents to solving equations involving such expressions. They reason through to the solution step by step. They explore extraneous solutions and explain why they occur.
WHAT: Students solve a radical equation to find when two runners in a race are in the same position. The resulting quadratic equation has one solution is relevant to the context. The other solution is not relevant because it occurs after the race is over. For this context, students can imagine that if the same equations continued to model Alice and Briana's race for an additional 5 kilometers, then they would meet again just over a half hour into the longer race.
WHY: The context in this task is “thin,” that is, a context that is not necessarily one that would arise in practice but that serves a mathematical purpose: to remind students that solving an equation is about finding when two expressions have the same numerical value. This can be further reinforced by graphing the two expressions on each side of the equation and interpreting the graphs in terms of the race. The task also introduces students to a function involving a square root, which grows more slowly than a linear function in the long run.
WHAT: In this task, students start by solving two similar looking radical equations. An extraneous solution arises in one process but not in the other. They then explore simpler cases and construct an argument for why the extraneous solution arises.
WHY: The purpose of the task is to show students a situation where squaring both sides of an equation results in an equation with solutions that satisfy the original equation. Though the reason for this may be unclear initially, students need to persevere in the problem solving process to be able to explain how and why the discrepancy exists (MP.1, MP.3).