The reciprocal numbers 1/α and 1/β are the roots; evaluate the exponential product.

01 / Standard#Your turn

2x^2+2x-1=0,\quad 2^{\alpha+1}2^{\beta+1}

Official paper · jm01-2025 · I.2 · PDF 2

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use Vieta for the reciprocal roots.

Hint 2

Their sum divided by their product equals α+β.

Worked solution

Recover the sum of α and β.
$\frac1\alpha+\frac1\beta=-1,\quad\frac1{\alpha\beta}=-\frac12\implies\alpha+\beta=2$
Add exponents with the same base.
$2^{\alpha+1}2^{\beta+1}=2^{\alpha+\beta+2}=16$

D: 16.

Checks and common pitfalls: α and β themselves are not the roots of the printed quadratic.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Recover the sum of α and β.
$\frac1\alpha+\frac1\beta=-1,\quad\frac1{\alpha\beta}=-\frac12\implies\alpha+\beta=2$
Add exponents with the same base.
$2^{\alpha+1}2^{\beta+1}=2^{\alpha+\beta+2}=16$

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

2025 JM01