LEARN · EXPLAIN · REVISE

Solve the linear–quadratic system completely.

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

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

TOPIC 01

Radicals, rational equations and simultaneous equations

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

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

01 / Standard#Your turn

y=x+3;\quad y=x^2

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

Write the domain first, then use reversible algebra and check the original equation.
Calculate or simplify this relation.
$x^2-x-3=0;\quad x=\frac{1\pm\sqrt{13}}{2}$
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

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

Teacher preparation and assessment

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?

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.

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.