Problem 3 · 2015 Math Kangaroo
Easy
Algebra & Patterns
substitution
How many real solutions does the equation 22x = 4x+1 have?
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Answer: A — \(0\)
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Hint 1 of 2
Write both sides as powers of the same base 2.
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Hint 2 of 2
Compare the exponents once the bases match.
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Approach: match bases, then compare exponents
- 4^(x+1) = (2²)^(x+1) = 2^(2x+2), so the equation is 2^(2x) = 2^(2x+2).
- Equal powers of 2 force 2x = 2x+2, i.e. 0 = 2, which is impossible.
- There are 0 real solutions (A).
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