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
- Use the discriminant as a condition on a parameter.
- Justify the conclusion "k=4 and the root is x=2" using the stated assumptions.
Before you start
Quadratic formula.
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
For x²−4x+k=0, which k gives exactly one distinct real solution?
Why this math matters
Use the discriminant as a condition on a parameter. 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 coefficient of x² is one and nonzero.
- Roots are counted as distinct real values.
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.
Determine when a quadratic has a repeated real root
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For x²−4x+k=0, which k gives exactly one distinct real solution?
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 discriminant is (−4)²−4k=16−4k
Its sign controls the real-root pattern.
Work through the mathematics
A repeated real root requires 16−4k=0
Zero discriminant makes the two quadratic-formula branches coincide.
Check the conclusion
k=4 and the root is x=2
The polynomial becomes (x−2)², showing the repeated factor directly.
The result
k=4 and the root is x=2
The polynomial becomes (x−2)², showing the repeated factor directly.
Common mistakes to catch
- One distinct root can have multiplicity two.
- A negative discriminant rules out real roots, not complex roots.
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
What happens when k>4?
Show a hint
The discriminant becomes negative.
Reveal answer and explanation
There are no real roots
The graph's minimum lies above zero.
Practice 2
What happens when k<4?
Show a hint
The discriminant is positive.
Reveal answer and explanation
Two distinct real roots
The square-root term separates the two solutions.
Take the idea with you
Use a parameter inequality to predict whether a model reaches a target level.
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: Place a rational number between two different real numbers
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