Whole Earth Curriculum WEC

Lesson 03 — Radical Equations: Justifying Every Step

Learners **solve radical equations** such as √(x+1) = 3 and √(2x−3) = x−3 by isolating the radical, squaring both sides, and **justifying each step with a property**. They discover that squaring can introduce an **extraneous solution**, and that the final check is not optional — it is part of honest mathematics.

D05 P3: Intellectual & Cognitive Awareness D05.S1 55 minutes Draft

Objectives

Essential question

How do I solve equations with radicals, and how do I check that a "solution" is real?

radical equationisolatesquare both sidesextraneous solutioninverse operation

Materials

Standard materials

Low-tech / no-cost

Enriched / lab & device

Works in different contexts

A two-column worked solution of the equation the square root of 2x minus 3 equals x minus 3, with each step listed on the left and the property that justifies it on the right, ending with a check that marks x equals 6 valid and x equals 2 extraneous
A two-column worked solution of the equation the square root of 2x minus 3 equals x minus 3, with each step listed on the left and the property that justifies it on the right, ending with a check that marks x equals 6 valid and x equals 2 extraneous

Lesson 3 — Radical Equations: Justifying Every Step

Summary

Learners solve radical equations such as √(x+1) = 3 and √(2x−3) = x−3 by isolating the radical, squaring both sides, and justifying each step with a property. They discover that squaring can introduce an extraneous solution, and that the final check is not optional — it is part of honest mathematics.

Objectives

  • Work with real numbers, roots, and radicals in equations, explaining which properties justify each step. (D05.S1.09.01)

Connection

If someone told you a square garden plot has an area of 25 square meters and asked how long its side is, you would think “a number whose square is 25” — that is √25 = 5 meters. That tiny move — turning “the square is 25” into “the side is 5” — is exactly what solving a radical equation does, but in reverse: it undoes a square root. Today you learn to undo roots honestly, checking your answer instead of trusting it, because sometimes the algebra hands you a number that does not actually fit.

Materials

  • Justification-column sheet
  • Worked-example card

Preparation

  • Copy the justification-column sheet and worked-example card.
  • Retrieval: from Grade 8, recall how to solve linear equations and how squaring and taking the square root undo each other.

Facilitator note

This lesson is written to the learner (“you”). The skill to land: to solve a radical equation, (1) isolate the radical on one side; (2) square both sides (or raise to the power that matches the root) to undo it; (3) solve the resulting equation; (4) substitute every candidate answer back into the original equation, because squaring can create an extraneous solution — a number that solves the squared equation but not the original. This is a skill, so use a worked example first, then guided practice (S-011). Model √(x+1) = 3 first (clean), then √(2x−3) = x−3, which produces two candidates, x = 6 and x = 2, of which only x = 6 checks. The critical-thinking lens: the check is what separates “I did algebra” from “I know it is true.” The ethics lens: honesty about extraneous solutions is honesty about evidence — do not keep an answer just because the algebra produced it (philosophy §5). The global lens: solving by “undoing” and justifying steps with properties descends from the algebraic tradition named by al-Khwarizmi, whose title gave us the word algebra (from al-jabr, “the reuniting”); early arguments were carried by pictures as much as symbols (S-429, S-431). The egalitarian lens: checking every answer is a habit open to everyone — no “math gift” is required, only care; “not yet” — a step that needs fixing — is information, never a verdict (philosophy §8). The technology lens: a graphing tool can show why an extraneous solution does not appear on the graph, but it confirms the check — it does not replace the honest habit of substituting the answer back into the original equation.

Procedure

  1. Recall (5 min). From Grade 8: how do you undo addition? (subtract) Multiplication? (divide) A square? (take the square root). What undoes a square root? (squaring).
  2. Study the worked example (12 min). Solve √(x+1) = 3. Step 1: the radical is already isolated. Step 2: square both sides — x + 1 = 9. Step 3: subtract 1 — x = 8. Check: √(8+1) = √9 = 3. ✓
  3. Meet the trap (15 min). Solve √(2x−3) = x−3. Square both sides: 2x − 3 = (x−3)² = x² − 6x + 9. Rearrange to zero: x² − 8x + 12 = 0. Factor: (x−6)(x−2) = 0. Candidates: x = 6 or x = 2. Check each in the original. x = 6: √(12−3) = √9 = 3 and 6−3 = 3. ✓. x = 2: √(4−3) = √1 = 1, but 2−3 = −1. ✗. So x = 2 is extraneous — it appeared when we squared, but it does not solve the original. The answer is x = 6 only.
  4. Guided practice (15 min). With a partner, solve √(x+2) = 4 and √(x−1) = x−3 using the two-column sheet. Name the property at each step and check every answer.
  5. Peer-check (5 min). Swap sheets. Did each step have a property? Was every candidate checked in the original equation?
  6. Close (3 min). Squaring can open a door to false answers. Checking is not a formality — it is how you know.

Differentiation

  • Support: Begin with √x = 4 and √x = 9 (no rearrangement needed), then add one step at a time.
  • Extension: Solve √(x+3) + 2 = x and explain, in words, exactly where the extraneous solution appears.

Assessment

  • Formative (observation): Can the learner justify each step with a property and correctly reject the extraneous solution in √(2x−3) = x−3?
  • Portfolio artifact: One fully checked two-column solution, kept in the portfolio.

Home connection

Make up a radical equation for someone at home (start simple: √x = 5), have them solve it, then show them the check — and explain why checking matters.

Resources