Solve for x: 2^(2x) - 10 * 2^x + 16 = 0.

Answer and explanation

Correct answer: C. x = 1 and x = 3

  1. Step 1: Let u = 2^x. Then 2^(2x) = (2^x)^2 = u^2.
  2. Step 2: Rewrite the equation as a quadratic in u: u^2 - 10u + 16 = 0.
  3. Step 3: Factor: (u - 2)(u - 8) = 0.
  4. Step 4: This gives u = 2 or u = 8.
  5. Step 5: Back-substitute u = 2^x:
  6. Step 6: 2^x = 2 => x = 1.
  7. Step 7: 2^x = 8 = 2^3 => x = 3.
  8. Step 8: The solutions are x = 1 and x = 3.
Answer
x = 1 and x = 3.
Why not the others: 'x = 2 and x = 8' are the values for the dummy substitution variable u, not x; 'x = 4 and x = 1' comes from factoring errors; 'x = 0 and x = 4' fails verification (2^0 - 10(1) + 16 = 7 != 0).

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