Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Check transformed equation candidates in the original radical equation.
- Justify the conclusion "Only x=3 solves the original equation" using the stated assumptions.
Before you start
Square roots and quadratics.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Solve √(x+6)=x.
Why this math matters
Check transformed equation candidates in the original radical equation. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- All roots and unknowns are real.
- √ denotes the nonnegative square root.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Reject a root introduced by squaring
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Solve √(x+6)=x.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
The right side must be nonnegative, so x≥0
A principal square root cannot equal a negative value.
Work through the mathematics
Squaring gives x+6=x², or (x−3)(x+2)=0
The transformed equation has candidates three and minus two.
Check the conclusion
Only x=3 solves the original equation
√9=3 works, while √4=2 is not −2.
The result
Only x=3 solves the original equation
√9=3 works, while √4=2 is not −2.
Common mistakes to catch
- A valid squared equation is not always equivalent to the original.
- Domain restrictions apply before algebraic manipulation.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Solve √(x+2)=−1 over R.
Show a hint
A principal square root is nonnegative.
Reveal answer and explanation
No solution
Squaring would hide the impossible sign condition.
Practice 2
Why must a squared solution be checked?
Show a hint
Squaring maps opposite signs to the same value.
Reveal answer and explanation
The transformed equation may be less restrictive
It can introduce candidates that fail the original equality.
Take the idea with you
Explain how a non-injective operation can create extra algebraic candidates.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Solve an absolute-value equation with a quadratic inside
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