If x = 7 − 4√3, then what is the value of x + 1/x?

Introduction / Context:
This algebra question involves surds and reciprocals. We are given x = 7 − 4√3 and must compute x + 1/x. The problem tests skill in rationalizing expressions with square roots and recognizing patterns that lead to simplification.

Given Data / Assumptions:
- x = 7 − 4√3.
- We need the exact value of x + 1/x.
- All operations are in real numbers and √3 is the positive square root of 3.

Concept / Approach:
When x is of the form a − b√c, its reciprocal 1/x can often be found by multiplying numerator and denominator by the conjugate a + b√c. Then x + 1/x frequently simplifies to a rational number because the surd parts cancel. We use the fact that (a − b√c)(a + b√c) = a² − b²c.

Step-by-Step Solution:
Step 1: Start with x = 7 − 4√3.Step 2: To find 1/x, multiply numerator and denominator by the conjugate 7 + 4√3.Step 3: So 1/x = (7 + 4√3) / [(7 − 4√3)(7 + 4√3)].Step 4: Compute the denominator using a² − b²c: 7² − (4² * 3) = 49 − 16 * 3 = 49 − 48 = 1.Step 5: Therefore, 1/x = 7 + 4√3.Step 6: Now compute x + 1/x = (7 − 4√3) + (7 + 4√3).Step 7: The surd terms cancel: −4√3 + 4√3 = 0, leaving 7 + 7 = 14.

Verification / Alternative check:
We can compute x numerically. Since √3 is approximately 1.732, x ≈ 7 − 4 * 1.732 ≈ 7 − 6.928 ≈ 0.072. Then 1/x ≈ 1 / 0.072 ≈ 13.888. Their sum is approximately 13.96, which is close to 14, confirming the exact result with minor rounding error.

Why Other Options Are Wrong:
The options involving surds, such as 3√3, 8√3 or 14 + 8√3, would indicate that the surd term survived the simplification. Because of the conjugate property, the surd part cancels when we add x and 1/x, leaving a pure rational result. Thus those options do not match the correct algebraic simplification.

Common Pitfalls:
Students may forget to use the conjugate properly or may incorrectly compute a² − b²c in the denominator, leading to wrong values for 1/x. Another mistake is to try to add x and 1/x before simplifying 1/x, which makes the algebra more complicated and error prone.

Final Answer:
The value of x + 1/x is 14.