In an algebraic simplification problem, if 2x + 9/x = 9 for a real non zero number x, then what is the minimum possible value of the expression x^2 + 1/x^2?

Introduction / Context:
This aptitude question tests algebraic manipulation and the use of quadratic equations to find the minimum value of an expression in x and 1/x. Problems of this type are very common in bank exams, SSC exams, and other competitive tests because they combine equation solving with optimization in a single short statement.

Given Data / Assumptions:

2x + 9/x = 9 for a real non zero number x.
The required expression is x^2 + 1/x^2.
We assume x is real so that the minimum value is interpreted over the real numbers.

Concept / Approach:
The key idea is to clear the denominator to obtain a quadratic equation in x, solve for its real roots, and then evaluate x^2 + 1/x^2 for each root. Since there are only two possible real values of x, the minimum value of x^2 + 1/x^2 will be the smaller of the two corresponding values. This approach avoids unnecessary calculus and relies only on standard quadratic and algebraic identities.

Step-by-Step Solution:
Start from 2x + 9/x = 9.Multiply both sides by x (x is non zero): 2x^2 + 9 = 9x.Rearrange to get a quadratic: 2x^2 - 9x + 9 = 0.Compute the discriminant: D = (-9)^2 - 4 * 2 * 9 = 81 - 72 = 9.Roots are x = [9 ± 3] / (2 * 2) = (9 ± 3) / 4.So x = 12/4 = 3 or x = 6/4 = 3/2.For x = 3, x^2 + 1/x^2 = 9 + 1/9 = 82/9.For x = 3/2, x^2 + 1/x^2 = (9/4) + (4/9) = (81 + 16) / 36 = 97/36.Between 82/9 and 97/36, clearly 97/36 is smaller.

Verification / Alternative check:
We can also use the identity x^2 + 1/x^2 = (x - 1/x)^2 + 2, which shows the expression is always at least 2. Our computed value 97/36 is greater than 2 and therefore reasonable. Substituting x = 3 and x = 3/2 back into the original equation 2x + 9/x = 9 confirms that both satisfy the given condition, so no other real x values need to be considered.

Why Other Options Are Wrong:
Values like 95/36, 87/25, 623/27, and 25/4 do not arise from any real solution of the quadratic 2x^2 - 9x + 9 = 0. They either correspond to incorrect algebraic manipulation or to evaluating x^2 + 1/x^2 for values of x that do not satisfy the original equation. Among all valid candidates, 97/36 is strictly the smallest value.

Common Pitfalls:
Many learners forget to multiply the equation by x correctly and make sign or coefficient errors in the quadratic. Another frequent mistake is to assume that the first value found for x gives the minimum without checking the second root. Some students also try to square the whole equation unnecessarily, which creates extra work and potential extraneous solutions. Carefully forming the quadratic and evaluating both resulting values of x^2 + 1/x^2 avoids these errors.

Final Answer:
The minimum possible value of x^2 + 1/x^2, given that 2x + 9/x = 9, is 97/36.