If x + y + z = 0, then what is the value of (3y^2 + x^2 + z^2) ÷ (2y^2 - xz)?

Introduction / Context:
This algebraic simplification question uses a condition linking x, y and z to reduce a seemingly complicated rational expression to a simple constant. Such identities are common in competitive exams and test how well a student can manipulate expressions using given relations.

Given Data / Assumptions:

• x, y and z are real numbers.
• Condition: x + y + z = 0.
• We must evaluate (3y^2 + x^2 + z^2) ÷ (2y^2 - xz).

Concept / Approach:
The key is to use the condition x + y + z = 0 to express x + z in terms of y and then relate x^2 + z^2 and xz to y. By rewriting x^2 + z^2 using (x + z)^2 and xz, we can simplify the numerator and denominator in a systematic way. Such manipulations rely on standard algebraic identities like (x + z)^2 = x^2 + z^2 + 2xz.

Step-by-Step Solution:
Given x + y + z = 0, rearrange to x + z = -y. Square both sides: (x + z)^2 = (-y)^2. So x^2 + z^2 + 2xz = y^2. Hence x^2 + z^2 = y^2 - 2xz. Now rewrite the numerator: 3y^2 + x^2 + z^2 = 3y^2 + (y^2 - 2xz) = 4y^2 - 2xz. The denominator is 2y^2 - xz. Observe that 4y^2 - 2xz = 2(2y^2 - xz). Therefore, (3y^2 + x^2 + z^2) ÷ (2y^2 - xz) = [2(2y^2 - xz)] ÷ (2y^2 - xz) = 2.

Verification / Alternative check:
Choose any convenient numbers satisfying x + y + z = 0, for example x = 1, y = 2, z = -3. Compute numerator: 3y^2 + x^2 + z^2 = 3*(4) + 1 + 9 = 12 + 1 + 9 = 22. Compute denominator: 2y^2 - xz = 2*4 - (1 * -3) = 8 + 3 = 11. Then numerator / denominator = 22 / 11 = 2, which matches the derived value.

Why Other Options Are Wrong:
Values 1, 3/2, 5/3 and 4/3 would arise only if the relation x + y + z = 0 were not fully used or if the identity for x^2 + z^2 was applied incorrectly. Because the simplification leads exactly to 2, any other constant conflicts with both algebra and the numeric check.

Common Pitfalls:
Learners sometimes attempt to substitute z = -x - y directly into the expression and expand everything blindly, which can be done but is lengthy and error prone. Using structured identities like (x + z)^2 makes the work much cleaner. Another mistake is to forget that (-y)^2 = y^2, not -y^2.

Final Answer:
The value of the expression is 2.