Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Radicals, rational equations and simultaneous equations

Read the idea, work independently, then explain what changed.

2027 JM01 考試大綱 · 6, 9, 14(A) · PDF 2 / printed page 2

Revisit first: Properties of equalities and inequalitiesQuadratic functions, equations and inequalities

TOPIC 01

Radicals, rational equations and simultaneous equations

Build understanding of radicals, rational equations and simultaneous equations through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Choose an equivalent method and explain exclusions before calculating.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Domains first

Denominators are nonzero; radicands are nonnegative; squaring also needs a sign check.

Equivalence and substitution

Substitution connects a system; every candidate is checked in its original equations.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in radicals, rational equations and simultaneous equations changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

The difference is 2ab=4. Equality holds exactly when ab=0.(a+b)² = 9a²+b² = 5

The difference is 2ab=4. Equality holds exactly when ab=0.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

Simplify the radical.

72\sqrt{72}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
Extract a positive square factor.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    36⋅2=62\sqrt{36\cdot2}=6\sqrt2

  3. A square root denotes the nonnegative root.

6√2.

Checks and common pitfalls: A square root denotes the nonnegative root.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Solve the rational equation.

x+1x−2=2\frac{x+1}{x-2}=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
Multiply by x−2 after excluding x=2.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    x≠2;x+1=2x−4;x=5x\ne2;\quad x+1=2x-4;\quad x=5

  3. Check that the proposed root leaves a nonzero denominator.

The requested value is 5.

Checks and common pitfalls: Check that the proposed root leaves a nonzero denominator.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Explain why squaring both sides may introduce a root, using this equation.

x+2=−x\sqrt{x+2}=-x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
Keep the sign restriction on −x before squaring.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    x≤0;x2−x−2=0x\le0;\quad x^2-x-2=0

  3. The positive quadratic candidate makes the right side negative.

Only x=(1−√9)/2 satisfies the original equation.

Checks and common pitfalls: The positive quadratic candidate makes the right side negative.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Simplify the radical.

162\sqrt{162}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
Extract a positive square factor.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    81⋅2=92\sqrt{81\cdot2}=9\sqrt2

  3. A square root denotes the nonnegative root.

9√2.

Checks and common pitfalls: A square root denotes the nonnegative root.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Solve the rational equation.

x+1x−3=2\frac{x+1}{x-3}=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
Multiply by x−3 after excluding x=3.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    x≠3;x+1=2x−6;x=7x\ne3;\quad x+1=2x-6;\quad x=7

  3. Check that the proposed root leaves a nonzero denominator.

The requested value is 7.

Checks and common pitfalls: Check that the proposed root leaves a nonzero denominator.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Solve the radical equation and reject any extraneous candidate.

x+3=x\sqrt{x+3}=x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
The square root forces the right side to be nonnegative.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    x≥0;x2−x−3=0;x=1±132x\ge0;\quad x^2-x-3=0;\quad x=\frac{1\pm\sqrt{13}}{2}

  3. The negative quadratic root cannot satisfy the unsquared equation.

x=(1+√13)/2.

Checks and common pitfalls: The negative quadratic root cannot satisfy the unsquared equation.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

Solve the linear system; enter x.

x+y=6;2x−y=9x+y=6;\quad 2x-y=9
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
Add the equations to eliminate y.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    3x=15;x=5;y=13x=15;\quad x=5;\quad y=1

  3. A system requires the pair to satisfy both equations.

The requested value is 5.

Checks and common pitfalls: A system requires the pair to satisfy both equations.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Solve the linear–quadratic system completely.

y=x+3;y=x2y=x+3;\quad y=x^2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
Substitute the linear expression for y.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    x2−x−3=0;x=1±132x^2-x-3=0;\quad x=\frac{1\pm\sqrt{13}}{2}

  3. Both real roots give intersections; neither can be discarded merely for being negative.

x=(1±√13)/2; y=x+3.

Checks and common pitfalls: Both real roots give intersections; neither can be discarded merely for being negative.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Is cancelling x−3 sufficient to conclude that the original equation has every real solution? Explain.

x2−9x−3=x+3\frac{x^2-9}{x-3}=x+3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
A cancelled factor retains its original exclusion.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    (x−3)(x+3)x−3=x+3(x≠3)\frac{(x-3)(x+3)}{x-3}=x+3\quad(x\ne3)

  3. A simplified identity does not fill a hole in the original domain.

No: all real x except 3.

Checks and common pitfalls: A simplified identity does not fill a hole in the original domain.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Explain why squaring both sides may introduce a root, using this equation.

x+3=−x\sqrt{x+3}=-x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
Keep the sign restriction on −x before squaring.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    x≤0;x2−x−3=0x\le0;\quad x^2-x-3=0

  3. The positive quadratic candidate makes the right side negative.

Only x=(1−√13)/2 satisfies the original equation.

Checks and common pitfalls: The positive quadratic candidate makes the right side negative.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Simplify the radical.

288\sqrt{288}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
Extract a positive square factor.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    144⋅2=122\sqrt{144\cdot2}=12\sqrt2

  3. A square root denotes the nonnegative root.

12√2.

Checks and common pitfalls: A square root denotes the nonnegative root.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Solve the rational equation.

x+1x−4=2\frac{x+1}{x-4}=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
Multiply by x−4 after excluding x=4.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    x≠4;x+1=2x−8;x=9x\ne4;\quad x+1=2x-8;\quad x=9

  3. Check that the proposed root leaves a nonzero denominator.

The requested value is 9.

Checks and common pitfalls: Check that the proposed root leaves a nonzero denominator.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Solve the radical equation and reject any extraneous candidate.

x+4=x\sqrt{x+4}=x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the domain first, then use reversible algebra and check the original equation.
Hint 2
The square root forces the right side to be nonnegative.
Worked solution

  1. Write the domain first, then use reversible algebra and check the original equation.

  2. Calculate or simplify this relation.

    x≥0;x2−x−4=0;x=1±172x\ge0;\quad x^2-x-4=0;\quad x=\frac{1\pm\sqrt{17}}{2}

  3. The negative quadratic root cannot satisfy the unsquared equation.

x=(1+√17)/2.

Checks and common pitfalls: The negative quadratic root cannot satisfy the unsquared equation.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for radicals, rational equations and simultaneous equations?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Choose an equivalent method and explain exclusions before calculating.
    • Domains first: Denominators are nonzero; radicands are nonnegative; squaring also needs a sign check.
    • Equivalence and substitution: Substitution connects a system; every candidate is checked in its original equations.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Denominators are nonzero; radicands are nonnegative; squaring also needs a sign check.
    • Expected reasoning: Substitution connects a system; every candidate is checked in its original equations.
    • Expected correction: A candidate produced by algebra may violate the original domain or sign condition.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

    No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

    Curriculum and source notes ↗