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
- Translate an area relationship into a polynomial.
- Use factoring and the zero-product property.
- Check both the equation and the physical domain.
Before you start
Expand binomials, factor simple quadratics, and calculate rectangular area.
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
In a geometry sketch, a rectangular garden has area 40 m² and is 3 m longer than it is wide. What dimensions satisfy this model?
Why this math matters
An area constraint multiplies unknown lengths, producing a quadratic. Factoring turns a zero-valued polynomial into simpler possibilities. Context determines which algebraic solutions represent usable lengths.
Set up the model
A useful answer starts with clear assumptions:
- The sketch is a perfect rectangle with positive side lengths.
- The area and length difference are exact model values; this is not a construction specification.
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.
Recover rectangle dimensions by factoring
PausedQuestion: Start with the question. Paused.
Question
Start with the question
In a geometry sketch, a rectangular garden has area 40 m² and is 3 m longer than it is wide. What dimensions satisfy this model?
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.
Express both dimensions
width = w; length = w + 3; w(w + 3) = 40
Using one variable automatically preserves the three-metre difference. The product of the lengths is the area.
Move to a zero product
w² + 3w − 40 = 0; (w + 8)(w − 5) = 0
Eight and negative five multiply to negative forty and add to three. Expanding the factors checks the polynomial.
Select and verify a root
w = −8 or 5; choose w = 5; length = 8 m
Negative eight is an algebraic root but violates positive width. The remaining dimensions satisfy 5 × 8 = 40 and 8 − 5 = 3.
The result
The sketch's dimensions are 5 m wide by 8 m long.
The zero-product property applies only after one side equals zero. You cannot set w or w + 3 equal to zero in the original product w(w + 3) = 40.
Common mistakes to catch
- Factoring the wrong sign on the constant term produces a different quadratic.
- A rejected contextual root should still be identified as an algebraic solution before explaining the domain restriction.
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
A rectangle has area 24 m² and length 2 m more than width. Find positive dimensions.
Show a hint
Factor w² + 2w − 24.
Reveal answer and explanation
4 m by 6 m
(w + 6)(w − 4) = 0 gives −6 or 4. The positive width is four and its length is six.
Practice 2
Solve x² − 9 = 0 over all real numbers.
Show a hint
Use a difference of squares; no length restriction is stated.
Reveal answer and explanation
x = −3 or x = 3
(x − 3)(x + 3) = 0 gives both roots. Each squares to nine, and both belong to the specified domain.
Take the idea with you
Separate the algebraic solution set from the contextually allowed solutions. Explain why any candidate is excluded.
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: Find dimensions when factoring is not convenient
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