Cogito
Integrated Math 3 · Chapter 3 · Lesson 2
Solving Radical Equations
Squaring can invent solutions.
12 problems · about 21 minutes · A-REI.A.2
What this lesson teaches
The student solves radical equations and rejects extraneous solutions.
- Isolate the radical, then raise both sides to the matching power.
- Squaring is not reversible, so extraneous solutions can appear.
- Check every candidate in the original equation.
Warm Up
Straightforward practice. Get the method working first.
5 problems√(x − 2) = 5. What is x?
Answer 27
Why 27.
Why can squaring introduce extraneous solutions?
Answer Squaring destroys the sign, so −3 and 3 become the same.
Why Squaring is not a reversible step.
√x = 7. What is x?
Answer 49
Why Square it.
√(x − 3) = 4. What is x?
Answer 19
Why x − 3 = 16.
√x = −2. How many solutions?
Answer 0
Why A square root is never negative.
Build It Up
The same ideas with more to keep track of.
3 problemsThe cube root of x = 2. What is x?
Answer 8
Why Cube both sides.
Do cube root equations produce extraneous solutions? 1 yes, 0 no.
Answer 0
Why Cubing is reversible.
√(2x) = 6. What is x?
Answer 18
Why 2x = 36.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the solving process in order.
Answer 1. Isolate the radical on one side. 2. Raise both sides to the matching power. 3. Solve the resulting equation. 4. Check every candidate in the original and reject failures.
Why Isolating comes before squaring.
√(x + 1) = 3. What is x?
Answer 8
Why x + 1 = 9.
The Solve: √x = 5. What is x?
Answer 25
Why 25.
The Rejection: √(x + 6) = x. How many valid solutions?
Answer 1
Why 1.