AMC 8 · 2019 · #20
Grade 6 algebraPick an answer.
AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Whenever we see (something)² = 16, the "something" is forced to be either +4 or -4 — the only two real numbers whose square is 16. That turns one scary quartic into two friendly questions: "when is x² - 5 = 4?" and "when is x² - 5 = -4?" Tool #6 (Guess and Check) lets us scan small integers x = 0, ± 1, ± 2, ± 3, ± 4 to spot every value where the inside hits 9 or 1. Tool #2 (Systematic List) keeps the scan organized so nothing is missed, and Tool #3 (Eliminate) confirms the count matches choice (D) and rules out the other choices.
Undo the outer square: (x² - 5)² = 16 holds exactly when the inside x² - 5 is +4 or -4.
Knowing that only 4 and -4 square to 16 uses the meaning of the exponent 2 — a Grade 6 idea.
6.EE.A.1Eliminate PossibilitiesAdd 5 to both sides of each branch, turning them into x² = 9 and x² = 1.
Splitting one equation into two easier ones is the Grade 6 "solve an equation of the form p x = q" pattern.
6.EE.B.7Identify SubproblemsScan small integers and their negatives for x² = 9 — the hits are x = 3 and x = -3.
Trying both +3 and -3 on a number line is exactly what Grade 6 "positive and negative numbers" expects.
6.NS.C.6Guess And CheckDo the same scan for x² = 1 — the hits are x = 1 and x = -1.
Reusing the squared-integer table avoids extra work and reinforces that (-1)² = 1 just like 1² = 1.
6.NS.C.6Guess And CheckNothing with |x| ≥ 4 or hidden between integers works, so the complete list is {-3, -1, 1, 3}.
Listing every value that satisfies the equation in order is the Grade 6 "find all values that make an equation true" idea.
6.EE.B.5Make A Systematic ListThis AMC 8 problem only needs Grade 6 exponents and the rule that both +a and -a square to a² — concepts you already know!