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Algebra I · 35 minutes

Suggested grades 9–10 · Unit 3: Quadratics and discrete patterns · Lesson 1 of 2

Quadratic Roots and a Frame Design

Use factoring and a contextual domain to choose a meaningful quadratic solution.

Mr. S Teaches · Original starter lesson · HS-ALG1-05 · Grade range is planning guidance.

Watch, notice, discuss

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After watching

Why can an algebraically correct root still be rejected in a measurement problem?

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Learning objective

Solve a factorable quadratic and reject a root that violates the stated measurement domain.

Before you begin

  • Factoring trinomials
  • Zero-product property

Gather your materials

  • Paper and pencil
  • Graph paper
  • Calculator for checking results

Language to know

root
An input making a polynomial equal zero.
zero-product property
If a product is zero, at least one factor must be zero.
domain restriction
A condition on allowed inputs.

Understand the concept

A rectangle’s sides are x and x+3, measured in centimeters. Requiring area 40 gives x²+3x−40=0. Factoring changes the equation’s form without changing its roots. Both algebraic roots must be checked, but a negative side length cannot represent the intended rectangle. Factoring does not solve every quadratic conveniently.

Teach it step by step

  1. Translate the rectangle’s area into x(x+3)=40 and rearrange to zero.
  2. Find factor constants multiplying to −40 and summing to 3.
  3. Solve each linear factor equation and verify the roots in the polynomial.
  4. Apply x>0 to the design and explain why contextual rejection differs from an algebra mistake.

Make the thinking visible

Worked example or model response

(x+8)(x−5)=0 gives x=−8 or 5. Both make x²+3x−40 zero; only x=5 gives positive dimensions 5 cm and 8 cm and area 40 cm².

Guided practice

Solve x²−x−12=0 and check both roots.

Independent practice

1. Solve x²+2x−15=0. 2. A rectangle has width w>0, length w+1 and area 20. Find its dimensions.

Quick assessment

Can (x−2)(x+5)=7 be solved by setting each factor to zero?

Answer key & assessment criteria

Answers and teaching guidance

  1. Guided: (x−4)(x+3)=0; roots 4 and −3.
  2. Independent 1: Roots 3 and −5. 2: w²+w−20=(w+5)(w−4), so width 4 and length 5.
  3. Quick check: No. The zero-product property applies to a product equal to zero, not seven.

Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.

Make it work for your learner

More support

Use an area box or factor-pair chart and keep algebraic roots separate from the context check.

Extension challenge

Compare a square-root method with factoring for x²−49=0.

Verified references · Selected contributions

Standards connections

Subject-content standards

These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.

Contributes to Common Core standard

HSA.REI.B.4.b

Common Core · High school · Algebra

Regular high school mathematics standard

Solve factorable quadratics and check their solutions.

Read the official standard ↗

Factoring and simple square-root comparison are selected methods; the full range of methods and complex roots require more instruction.

Verified 2026-11-04. Selected contribution, not full attainment.