A number is greater than twice its reciprocal by 31/4. What is the larger of the possible values of this number?

Introduction / Context:
This question involves a number and its reciprocal linked by a linear relationship. The statement says that the number is greater than twice its reciprocal by 31/4. From this, you are asked to find the larger of the possible values of the number. This leads to a quadratic equation and tests your ability to handle algebra involving reciprocals, fractions and multiple solutions.

Given Data / Assumptions:

- Let the non-zero number be x. - Its reciprocal is 1 / x. - The number x is greater than 2 * (1 / x) by 31 / 4. - This means x - 2 / x = 31 / 4. - We must find the larger of the possible values of x.

Concept / Approach:
The given relation directly translates into an equation that we can manipulate into a quadratic. Multiplying through by x clears the denominator. After rearranging into standard quadratic form, we solve using the quadratic formula or factoring. This will usually produce two real solutions. Because the question explicitly asks for the larger value, we will compare the solutions and report the greater one. Throughout, it is important to remember that x must be non zero so that the reciprocal is defined.

Step-by-Step Solution:
Step 1: Translate the relationship into an equation: x - 2 / x = 31 / 4. Step 2: Multiply both sides by x to eliminate the denominator: x^2 - 2 = (31 / 4) * x. Step 3: Multiply both sides by 4 to clear the fraction: 4x^2 - 8 = 31x. Step 4: Rearrange to standard quadratic form: 4x^2 - 31x - 8 = 0. Step 5: Identify coefficients: a = 4, b = -31, c = -8. Step 6: Compute the discriminant D = b^2 - 4ac = (-31)^2 - 4 * 4 * (-8) = 961 + 128 = 1089. Step 7: The square root of 1089 is 33, since 33 * 33 = 1089. Step 8: Use the quadratic formula: x = [31 ± 33] / (2 * 4) = [31 ± 33] / 8. Step 9: First root: x = (31 + 33) / 8 = 64 / 8 = 8. Step 10: Second root: x = (31 - 33) / 8 = -2 / 8 = -1 / 4. Step 11: The two possible values of x are 8 and -1 / 4, and the larger of these is 8.

Verification / Alternative check:
Verify the relation for x = 8. The reciprocal is 1 / 8. Compute twice the reciprocal: 2 * (1 / 8) = 1 / 4. Then x - 2 / x becomes 8 - 1 / 4 = (32 / 4) - (1 / 4) = 31 / 4, exactly as given in the problem statement. This confirms that x = 8 satisfies the condition. If desired, you can also check x = -1 / 4, which also satisfies the equation, but is smaller than 8 and therefore not the answer requested.

Why Other Options Are Wrong:
Values 6, 7 and 9 do not satisfy the relationship when substituted into x - 2 / x. For example, with x = 7, we get 7 - 2 / 7, which is far from 31 / 4. These values might look attractive as simple integers but do not arise from solving the quadratic equation correctly. Only 8 is the larger solution produced by the correct algebraic process.

Common Pitfalls:
Common errors include forgetting to multiply the entire equation by x or 4 when clearing denominators, which leads to an incomplete or incorrect quadratic. Some students may not consider that a quadratic can have two real solutions and thus fail to check both, or they might misinterpret the phrase "greater than" when forming the equation. Writing each step explicitly, including the non zero restriction, and verifying solutions in the original relation helps avoid these issues.

Final Answer:
The larger possible value of the number is 8, which corresponds to option B.