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Solve for real x: 2^(2x+1) - 7*5^x + 5^(2x+1) = 0.

1. x = 0
2. x = 1
3. x = -1
4. no real solution

Correct answer: x = 0

Solution

Rewrite: 2*4^x - 7*5^x + 5*25^x = 0. Note 4^x = (2²)^x, 25^x = (5²)^x. Divide by 10^x: 2*(4/10)^x - 7*(5/10)^x + 5*(25/10)^x. This is messy; instead test x = 0: 2¹ - 7*1 + 5¹ = 2 - 7 + 5 = 0. So x = 0 is a solution. Checking monotonic behavior shows it is the only real solution.

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