Find a,b,c so that every solution of the preceding linear system also satisfies the quadratic constraint.

01 / Standard#Your turn

ax^2+by^2+cz^2=3

Official paper · jm02-2026 · 5(b)(ii) · PDF 7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Substitute the full parameterised solution.

Hint 2

The resulting polynomial must be an identity for all real t.

Worked solution

Expand after substitution.
$a(1+t)^2+b(1-2t)^2+ct^2=(a+4b+c)t^2+(2a-4b)t+(a+b)$
Match coefficients with the constant 3 and solve.
$a+4b+c=0,\quad2a-4b=0,\quad a+b=3\implies(a,b,c)=(2,1,-6)$

a=2, b=1, c=−6.

Checks and common pitfalls: Checking one parameter value is insufficient when every solution is required.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Expand after substitution.
$a(1+t)^2+b(1-2t)^2+ct^2=(a+4b+c)t^2+(2a-4b)t+(a+b)$
Match coefficients with the constant 3 and solve.
$a+4b+c=0,\quad2a-4b=0,\quad a+b=3\implies(a,b,c)=(2,1,-6)$

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

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