If x^2 – 12x + 33 = 0, then what is the value of (x – 4)^2 + 1/(x – 4)^2 in algebraic simplification?

Introduction / Context:
This question tests algebraic manipulation of a quadratic equation and the ability to transform it into a convenient form in order to evaluate an expression involving x minus a constant. Such problems are common in aptitude exams because they check whether you can use the relationship given by an equation to compute new expressions without first finding full decimal values of the roots.

Given Data / Assumptions:

• x satisfies the quadratic equation x^2 - 12x + 33 = 0.
• The required value is (x - 4)^2 + 1 / (x - 4)^2.
• x is assumed to be a real number that satisfies the equation.

Concept / Approach:
The key idea is to rewrite the quadratic equation in terms of a shifted variable t = x - 4 so that the required expression becomes t^2 + 1 / t^2. Once we have an equation in t, we can find t + 1 / t and then use the identity t^2 + 1 / t^2 = (t + 1 / t)^2 - 2. This avoids solving the quadratic in terms of square roots and keeps the work simpler and faster.

Step-by-Step Solution:
Let t = x - 4, so x = t + 4. Substitute x = t + 4 into x^2 - 12x + 33 = 0. We get (t + 4)^2 - 12(t + 4) + 33 = 0. Expand: t^2 + 8t + 16 - 12t - 48 + 33 = 0. Combine like terms: t^2 - 4t + 1 = 0. Rearrange: t^2 - 4t + 1 = 0 implies t^2 + 1 = 4t. Divide both sides by t (t is non zero because otherwise x - 4 would be zero and would not satisfy the equation): t + 1 / t = 4. Use the identity t^2 + 1 / t^2 = (t + 1 / t)^2 - 2. Compute: t^2 + 1 / t^2 = 4^2 - 2 = 16 - 2 = 14. Since t = x - 4, the required expression (x - 4)^2 + 1 / (x - 4)^2 equals t^2 + 1 / t^2 = 14.

Verification / Alternative check:
We can verify by solving x^2 - 12x + 33 = 0 explicitly. The roots are x = 6 ± sqrt(3). For x = 6 + sqrt(3), x - 4 = 2 + sqrt(3). Evaluating (2 + sqrt(3))^2 + 1 / (2 + sqrt(3))^2 simplifies to 14. For x = 6 - sqrt(3), x - 4 = 2 - sqrt(3), and the same expression again simplifies to 14. Thus the value is independent of which root is chosen.

Why Other Options Are Wrong:
16: This arises if someone takes t + 1 / t = 4 and directly uses 4^2 without subtracting 2, ignoring the correct identity.
18: This may appear if the identity is misapplied as t^2 + 1 / t^2 = (t + 1 / t)^2 + 2 instead of subtracting 2.
20: This is larger than the correct result and usually comes from algebraic mistakes in expansion.
12: This is smaller than the correct value and can come from incorrectly forming the equation in t or errors in combining like terms.

Common Pitfalls:
A common error is to try to find the numerical roots first, which introduces unnecessary complexity. Errors also occur when shifting from x to t and expanding the square incorrectly. Another mistake is forgetting the exact form of the identity for t^2 + 1 / t^2 or applying it with the wrong sign. Keeping the steps systematic and using identities carefully prevents these issues.

Final Answer:
The value of (x - 4)^2 + 1 / (x - 4)^2 is 14.