A number is greater than 58 times its reciprocal by 3/4. If x is the number, then x − 58/x = 3/4. What is the value of x?

Introduction / Context:
This algebraic problem involves a number and its reciprocal. We are told that the number exceeds 58 times its reciprocal by 3/4. Translating this into an equation in x leads to a quadratic equation, which we can solve to find the value(s) of the number.

Given Data / Assumptions:
Let the number be x (x ≠ 0).The number is greater than 58 times its reciprocal by 3/4.This means x − 58/x = 3/4.We must solve for x and choose the correct value from the options.

Concept / Approach:
The equation x − 58/x = 3/4 contains both x and 1/x. To clear the fraction in the denominator, we multiply both sides by x, converting the equation into a quadratic form. Then we clear the remaining fraction by multiplying through by 4 and solve the quadratic equation using the discriminant method.

Step-by-Step Solution:
Step 1: Start with x − 58/x = 3/4.Step 2: Multiply both sides by x (x ≠ 0) to eliminate the denominator: x^2 − 58 = (3/4) x.Step 3: Multiply both sides by 4 to clear the fractional coefficient: 4x^2 − 232 = 3x.Step 4: Bring all terms to one side: 4x^2 − 3x − 232 = 0.Step 5: This is a quadratic equation with a = 4, b = −3 and c = −232.Step 6: Compute the discriminant D = b^2 − 4ac = (−3)^2 − 4 × 4 × (−232) = 9 + 3712 = 3721.Step 7: Note that 61^2 = 3721, so √D = 61.Step 8: Solve for x: x = [3 ± 61] / (2 × 4) = (3 ± 61) / 8.Step 9: The two roots are x = (3 + 61) / 8 = 64 / 8 = 8 and x = (3 − 61) / 8 = −58 / 8 = −29 / 4.

Verification / Alternative check:
Check x = 8 in the original equation. The reciprocal is 1/8, so 58 times the reciprocal is 58 × 1/8 = 7.25. Then x − 58/x = 8 − 7.25 = 0.75 = 3/4, which satisfies the condition. The other root, −29/4, is not among the answer options. The closest option is −8 or 12, but neither satisfies the original equation.

Why Other Options Are Wrong:
Substituting x = 12 gives 12 − 58/12, which is not equal to 3/4. Similarly, x = −8 or x = −12 do not satisfy the equation x − 58/x = 3/4. Only x = 8 fulfills the relationship exactly.

Common Pitfalls:
Typical errors include misinterpreting the phrase “greater than 58 times its reciprocal by 3/4” (which must be written as x − 58/x = 3/4), or making algebraic mistakes when clearing denominators and fractions. Another common issue is miscomputing the discriminant or its square root.

Final Answer:
The number that satisfies the condition is 8.