AMC 10 · 2025 · #3
Grade 11 algebraPick an answer.
Multiplying the four factors left to right works, but it drags a messy complex number through three separate multiplications and gives three chances to lose a sign. So I rearrange the factors before touching them. The four factors are i shifted down by 0, 1, 2, and 3, and if I pair the outer two together and the inner two together, each pair's shifts add to the same total 3 — which forces both pairs into the same shape. That regrouping turns one long product into two short independent ones, and their matching shape then collapses in a single move. Tracking whether any i can survive to the end also crosses off most of the choices before the arithmetic is even finished.
Pair the outer and inner factors
Pair the outer two and the inner two.
Pairing the factors whose shifts add to the same total forces both pairs into the same form, and matching forms combine cleanly.
9.A-SSE.A.2Organize Information In More WaysMultiply each pair on its own
They become negative one minus three i and one minus three i.
Complex multiplication is ordinary expanding with one extra rule: every i² turns into -1.
11.N-CN.A.2Identify SubproblemsReveal a difference of squares
Pulling out a sign reveals a difference of squares.
When two factors differ only in the sign of one term, their middle terms cancel and only the squares survive.
When two factors differ only in the sign of one term, their middle terms cancel and only the squares survive.
▸ Why?
Multiplying a sum by its matching difference gives one square minus the other.
▸ Why?
Expanding spreads each term across the other, which is exactly what makes the middle terms pair off.
Check that no stray i survives
Every imaginary part cancels, leaving a real number.
A complex number is real exactly when no first-power i is left standing, and the cancellation removed all of them.
11.N-CN.A.1Eliminate PossibilitiesSubstitute and finish
Substituting gives -10.
Subtracting 9i² is the same as adding 9, so the bracket grows to 10 and the outer minus makes the result negative.
11.N-CN.A.2Identify SubproblemsBefore multiplying a long chain of factors, hunt for a pairing that makes the pieces match — matching pieces cancel each other's messy parts and leave a clean number behind.
- Pair the outer and inner factors
- Multiply each pair on its own
- Reveal a difference of squares
- Check that no stray i survives
- Substitute and finish