Solve for the missing radicand — In the equation 392 / sqrt(x) = 28, determine the value of x. Show each manipulation of the equation clearly.

Introduction / Context:This question checks your fluency with algebraic manipulation involving square roots and simple fractions. The structure 392 / sqrt(x) = 28 invites you to isolate the radical term and then square appropriately to solve for x without introducing extraneous solutions.

Given Data / Assumptions:

• Equation: 392 / sqrt(x) = 28.
• sqrt(x) denotes the principal (non-negative) square root.
• x is positive so that sqrt(x) is defined in real numbers.

Concept / Approach:Isolate sqrt(x) by multiplying both sides by sqrt(x) and dividing by 28, or equivalently by cross-multiplication. Once sqrt(x) is expressed as a number, square both sides to remove the radical and obtain x. Always verify by substituting back into the original equation to avoid sign or arithmetic slips.

Step-by-Step Solution:Start with 392 / sqrt(x) = 28.Rearrange: sqrt(x) = 392 / 28.Compute 392 / 28 = 14.Square both sides: x = 14^2 = 196.

Verification / Alternative check:Substitute x = 196: sqrt(196) = 14, and 392 / 14 = 28. The original equation is satisfied exactly, confirming the solution.

Why Other Options Are Wrong:

• 144: Would imply sqrt(x) = 12, giving 392 / 12 = 32.666..., not 28.
• 24 / 48: These are not valid values of x in this context; they would make sqrt(x) non-integer and yield a non-28 quotient.
• 784: sqrt(784) = 28 ⇒ 392 / 28 = 14, not 28.

Common Pitfalls:Squaring the entire original fraction prematurely; inverting 392/28 incorrectly; forgetting that sqrt(x) must be non-negative in the real-number context.

Final Answer:196