Solve the exponential equation 3^(x + 2) + 3^(−x) = 10 and identify which of the following ordered pair of real numbers correctly lists all solutions for x.

Introduction / Context:
This problem tests your ability to solve equations involving exponents with the same base, here base 3. By introducing a substitution, the equation can be turned into a quadratic in a single variable. This technique is common in algebra and aptitude tests whenever exponential expressions appear both as direct powers and as reciprocals.

Given Data / Assumptions:

• The equation is 3^(x + 2) + 3^(−x) = 10.
• x is a real number.
• We must determine all real solutions for x and then choose the option that lists them correctly.

Concept / Approach:
Because both terms use base 3, we can simplify by making a substitution such as t = 3^x. Then 3^(x + 2) becomes 9t and 3^(−x) becomes 1 / t. The original equation changes into an equation in t that is quadratic in form after we clear denominators. Solving that quadratic gives possible values for t, and from each t value we solve 3^x = t to find x. Finally, we check that the values satisfy the original equation.

Step-by-Step Solution:
Let t = 3^x. Then t > 0 for all real x. Compute 3^(x + 2) = 3^x * 3^2 = 9t. Compute 3^(−x) = 1 / 3^x = 1 / t. Substitute into the equation: 9t + 1 / t = 10. Multiply both sides by t (t > 0) to clear the denominator: 9t^2 + 1 = 10t. Rearrange to standard quadratic form: 9t^2 - 10t + 1 = 0. Solve using the quadratic formula: t = [10 ± sqrt(10^2 - 4 * 9 * 1)] / (2 * 9). The discriminant is 100 - 36 = 64, so t = [10 ± 8] / 18. Thus t = (10 + 8) / 18 = 18 / 18 = 1, or t = (10 - 8) / 18 = 2 / 18 = 1 / 9. Recall t = 3^x, so we solve 3^x = 1 and 3^x = 1 / 9. 3^x = 1 gives x = 0. 3^x = 1 / 9 = 3^(−2) gives x = -2.

Verification / Alternative check:
Check x = 0: 3^(0 + 2) + 3^(−0) = 3^2 + 3^0 = 9 + 1 = 10, which works. Check x = -2: 3^(−2 + 2) + 3^(2) = 3^0 + 3^2 = 1 + 9 = 10, which also works. No other t values came from the quadratic, and both t values were positive, so we have found all real solutions.

Why Other Options Are Wrong:
Pairs like 0 and 2, 1 and -1, or 1 and 2 arise from solving incorrect quadratics or misinterpreting the exponent rules. The pair -2 and 2 includes 2 as a solution, but x = 2 does not satisfy the equation, since 3^(2 + 2) + 3^(−2) = 81 + 1/9, which is far from 10. Only the pair 0 and -2 matches the correct solutions.

Common Pitfalls:
Mistakes often occur when clearing denominators or when using the quadratic formula. Some students incorrectly set t = 3^(−x) instead of 3^x and then mix up the transformations. Being careful with exponent rules (like 3^(x + 2) = 3^x * 9) and double checking the discriminant avoids these errors.

Final Answer:
The solutions of the equation are x = 0 and x = -2, so the correct option is 0 and -2.