In a quadratic equation, if x is a non zero real number that satisfies x^2 - 3x + 1 = 0, then what is the value of the expression x + 1/x?

Introduction / Context:
This question tests a standard algebra technique where we use the relationship between a root of a quadratic equation and its reciprocal. Instead of solving for x explicitly, we try to find x + 1/x directly from the given quadratic equation x^2 - 3x + 1 = 0.

Given Data / Assumptions:

• x is a non zero real number.
• x satisfies x^2 - 3x + 1 = 0.
• We must find the exact value of x + 1/x.

Concept / Approach:
The key idea is to use properties of quadratic equations and symmetric expressions. If x is a root of x^2 - 3x + 1 = 0, then we can manipulate the equation to obtain an expression for x + 1/x without computing x explicitly. Another way is to recognize that for such quadratics with constant term 1, the reciprocal 1/x is also a root, so the sum of the two roots equals x + 1/x.

Step-by-Step Solution:
Step 1: Start from the equation x^2 - 3x + 1 = 0. Step 2: Divide the entire equation by x (allowed because x is non zero): x - 3 + 1/x = 0. Step 3: Rearrange to isolate x + 1/x: x + 1/x = 3. Step 4: Therefore the required value of the expression x + 1/x is 3.

Verification / Alternative check:
We can also use the root and reciprocal idea. For the quadratic x^2 - 3x + 1 = 0, the sum of roots is 3 and the product of roots is 1. Because the product is 1, the second root must be the reciprocal of the first. Hence if one root is x, the other root is 1/x, so x + 1/x equals the sum of roots, which is 3. This confirms our result.

Why Other Options Are Wrong:
Option A (1) is wrong because it would require x - 3 + 1/x = -2, which contradicts the given equation. Option B (0) is impossible because it would give x + 1/x = 0 and would not satisfy x^2 - 3x + 1 = 0. Option D (2) is also inconsistent with the algebraic manipulation. Option E (4) is larger than the actual symmetric value that comes from the quadratic relation.

Common Pitfalls:
A frequent mistake is to try to solve the quadratic equation completely, introduce square roots, and then compute x + 1/x numerically. This is longer and more error prone. Another common error is dividing incorrectly or forgetting that x must be non zero when dividing by x. Some learners also confuse x + 1/x with (x - 1/x) and apply the wrong identity.

Final Answer:

Thus, by directly manipulating the quadratic equation, the value of the expression is 3.