Learn with Amar
Teaching video
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Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Separate the two possible signs and reject impossible squared values.
- Justify the conclusion "The real solutions are x=−3 and x=3" using the stated assumptions.
Before you start
Absolute values and square roots.
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 |x²−4|=5.
Why this math matters
Separate the two possible signs and reject impossible squared values. 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:
- The solution set is sought over R.
- Absolute value has its real-number meaning.
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.
Solve an absolute-value equation with a quadratic inside
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Solve |x²−4|=5.
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
x²−4=5 or x²−4=−5
An absolute value of five permits either signed interior value.
Work through the mathematics
The first case gives x²=9; the second gives x²=−1
Each branch must be solved in the specified number system.
Check the conclusion
The real solutions are x=−3 and x=3
The negative-square branch supplies no real solution, and both remaining values check.
The result
The real solutions are x=−3 and x=3
The negative-square branch supplies no real solution, and both remaining values check.
Common mistakes to catch
- The negative branch may be impossible over the reals.
- Do not discard the negative square root when x² is positive.
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 |x²−4|=0.
Show a hint
The interior must vanish.
Reveal answer and explanation
x=−2 or 2
Zero has only one signed branch.
Practice 2
Would x=i satisfy the corresponding complex-modulus equation?
Show a hint
Substitute i²=−1 before taking the modulus.
Reveal answer and explanation
Yes
|i²−4|=|−5|=5, but complex modulus does not restrict the interior to only the two real values ±5.
Take the idea with you
Use case splitting to analyze equal-distance conditions involving a nonlinear measurement.
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: Combine logarithms without admitting forbidden inputs
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