Competition · AMC preparation · step 4 of 4
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
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.
The equation (x² - 5)² = 16 is satisfied by exactly the same real numbers x as the two simpler equations x² - 5 = 4 and x² - 5 = -4 taken together.
▸ Why?
A square can equal 16 only when the quantity being squared is itself a real number whose square is 16, and there are exactly two such numbers: 4 and its opposite -4.
▸ Why?
A positive number such as 16 has exactly two real square roots — one positive number and its negative — so the base x² - 5 is forced to be 4 or -4 and can be nothing else.
▸ Why?
Splitting the one equation into 'the =4 case or the =-4 case' keeps every solution and invents none, because the full set of solutions breaks with no gaps and no overlap into the part where x² - 5 = 4 and the part where x² - 5 = -4.
Simplify each case
Add 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 SubproblemsFind the squares that give 9
Scan 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 CheckFind the squares that give 1
Do 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 CheckCount all the solutions
Nothing 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!
- Undo the outer square
- Simplify each case
- Find the squares that give 9
- Find the squares that give 1
- Count all the solutions
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