Find all p for which the equation has a real solution.

01 / Standard#Your turn

px^2-2(p+3)x+p-1=0

Official paper · jm01-2022 · I.5 · PDF 2

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Treat p=0 separately.

Hint 2

For p≠0 require a nonnegative discriminant.

Worked solution

The quadratic discriminant simplifies to a linear expression.
$\Delta=4(p+3)^2-4p(p-1)=4(7p+9)\ge0\iff p\ge-9/7$
At p=0, the linear equation −6x−1=0 also has a real solution, already inside this range.

D: p≥−9/7.

Checks and common pitfalls: The question says equation, so the linear case must not be discarded.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The quadratic discriminant simplifies to a linear expression.
$\Delta=4(p+3)^2-4p(p-1)=4(7p+9)\ge0\iff p\ge-9/7$
At p=0, the linear equation −6x−1=0 also has a real solution, already inside this range.

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

2022 JM01