Find all real solutions of the radical equation.

01 / Standard#Your turn

\sqrt{3x^2-4x+1}-\sqrt{3x^2-2x+4}=-1

Official paper · jm01-2026 · I.11 · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Move one square root across before squaring.

Hint 2

After the first squaring, one root equals x+1.

Worked solution

Isolate and square, retaining the nonnegative-root condition.
$\sqrt{3x^2-2x+4}=\sqrt{3x^2-4x+1}+1\implies\sqrt{3x^2-4x+1}=x+1$
Square again and test both candidates in the original equation.
$3x^2-4x+1=(x+1)^2\implies2x(x-3)=0;\quad x=0:1-2=-1,\ x=3:4-5=-1$

D: {0,3}.

Checks and common pitfalls: Squaring gives candidates; substitution is still required.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Isolate and square, retaining the nonnegative-root condition.
$\sqrt{3x^2-2x+4}=\sqrt{3x^2-4x+1}+1\implies\sqrt{3x^2-4x+1}=x+1$
Square again and test both candidates in the original equation.
$3x^2-4x+1=(x+1)^2\implies2x(x-3)=0;\quad x=0:1-2=-1,\ x=3:4-5=-1$

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

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