Problem 20 · 2019 AMC 8
Medium
Algebra & Patterns
square-root-both-sidescasework
How many different real numbers x satisfy the equation
(x2 − 5)2 = 16 ?
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Answer: D — 4 real numbers.
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Hint 1 of 2
The question asks how many solutions — so peel the equation one square at a time. Something squared equals 16 means that something is +4 or −4. Don't lose the negative branch.
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Hint 2 of 2
Each branch leaves x2 = (a positive number), and a positive x2 always gives two values of x. Count the branches that stay positive.
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Approach: undo each square, keeping both signs
- (x2 − 5)2 = 16 means x2 − 5 = +4 or −4 — both, since either squares to 16.
- Branch +4: x2 = 9 ⇒ x = ±3 (two reals).
- Branch −4: x2 = 1 ⇒ x = ±1 (two more).
- Both branches gave a positive x2, so each yields 2 real roots: 4 total.
- Why this transfers: each square you undo can double the solution count — but only when the inside is positive (a negative x2 would give zero real roots). Track the ± at every layer.
Another way — difference of squares:
- Move 16 over: (x2 − 5)2 − 42 = 0, a difference of squares = (x2 − 5 + 4)(x2 − 5 − 4) = 0.
- That's (x2 − 1)(x2 − 9) = (x+1)(x−1)(x+3)(x−3) = 0 — four distinct linear factors, so 4 real roots.
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