The real roots x₁,x₂ of 3x²−8x+m=0 have reciprocal arithmetic mean 2. Find m.

01 / Standard#Your turn

Official paper · jm01-2022 · I.12 · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

The sum of reciprocals is 4.

Hint 2

Use root sum divided by root product.

Worked solution

Apply Vieta and the mean condition.
$\frac1{x_1}+\frac1{x_2}=\frac{8/3}{m/3}=\frac8m=4\implies m=2$
Check reality and nonzero roots.
$\Delta=64-12(2)=40>0,\quad x_1x_2=2/3\ne0$

E: m=2.

Checks and common pitfalls: Mean 2 means sum 4, not sum 2.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Apply Vieta and the mean condition.
$\frac1{x_1}+\frac1{x_2}=\frac{8/3}{m/3}=\frac8m=4\implies m=2$
Check reality and nonzero roots.
$\Delta=64-12(2)=40>0,\quad x_1x_2=2/3\ne0$

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

2022 JM01