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.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Identify quadratic coefficients and the discriminant.
- Apply the entire quadratic formula correctly.
- Distinguish exact roots from rounded measurements.
Before you start
Rearrange a quadratic equation, calculate square roots, and use parentheses on a calculator.
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
A rectangular display in a mathematical sketch has area 20 m² and length 2 m greater than its width. Find its positive dimensions without assuming whole-number lengths.
Why this math matters
Real-number solutions are not limited to tidy integers. The quadratic formula provides a systematic method when a polynomial does not factor conveniently over the integers. Exact radical forms keep calculations consistent until a rounded report is needed.
Set up the model
A useful answer starts with clear assumptions:
- The mathematical display is rectangular with positive lengths.
- Dimensions are model values, not a fabrication specification; final decimal lengths are rounded to two places.
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.
Find dimensions when factoring is not convenient
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A rectangular display in a mathematical sketch has area 20 m² and length 2 m greater than its width. Find its positive dimensions without assuming whole-number lengths.
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 quadratic
w(w + 2) = 20; w² + 2w − 20 = 0
The coefficients are a = 1, b = 2, and c = −20. Keep the negative sign attached to c.
Apply the formula
D = 2² − 4(1)(−20) = 84; w = (−2 ± √84)/2 = −1 ± √21
A positive discriminant gives two distinct real roots. The denominator divides both terms of the numerator.
Keep the positive dimensions
w = √21 − 1 ≈ 3.58 m; L = √21 + 1 ≈ 5.58 m
The other width is negative. Using exact forms checks the area: (√21 − 1)(√21 + 1) = 21 − 1 = 20.
The result
The exact dimensions are (√21 − 1) m and (√21 + 1) m, approximately 3.58 m by 5.58 m.
Multiplying the rounded decimals will give a value close to twenty rather than exactly twenty. That small difference comes from rounding, not from failure of the exact solution.
Common mistakes to catch
- Replacing c = −20 with positive twenty changes the discriminant.
- Rounding the square root early can compound errors in the recovered dimensions.
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 − 5x + 1 = 0 exactly.
Show a hint
Use a = 1, b = −5, c = 1.
Reveal answer and explanation
x = (5 ± √21)/2
The discriminant is 25 − 4 = 21. Applying the formula gives two real roots, both positive.
Practice 2
How many real roots does x² + 4x + 7 = 0 have?
Show a hint
Inspect the discriminant without completing the formula.
Reveal answer and explanation
No real roots
D = 16 − 28 = −12. A negative discriminant prevents real square roots, although complex solutions exist.
Take the idea with you
Keep exact expressions during verification, then round for communication at a precision appropriate to the input information.
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: Turn several kit orders into a supply list with matrices
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.