Find every m for which the expression is strictly positive for all real x.

01 / Standard#Your turn

mx^2+6x+3m

Official paper · jm01-2021 · I.4 · PDF 2

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Check the leading coefficient, including its zero case.

Hint 2

An upward quadratic must have negative discriminant to stay strictly positive.

Worked solution

For m<0 the tails are negative; for m=0 the expression 6x changes sign. Thus m>0 is necessary.
Exclude real roots and tangency to zero.
$\Delta=36-12m^2<0\iff m^2>3$
Combine the two restrictions.
$m>\sqrt3$

B: m>√3.

Checks and common pitfalls: A zero discriminant permits the value zero, so it cannot give strict positivity.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
For m<0 the tails are negative; for m=0 the expression 6x changes sign. Thus m>0 is necessary.
Exclude real roots and tangency to zero.
$\Delta=36-12m^2<0\iff m^2>3$
Combine the two restrictions.
$m>\sqrt3$

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

2021 JM01