AMC 10 · 2006 · #7

Grade 8 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

fraction-arithmeticabsolute-valuesigned-square-root identify-subproblems ↑ Prerequisites: fraction-arithmetic

📏 Medium solution 💡 2 insights

Problem

Given a number

x

with

x \lt 0

, simplify the expression

\sqrt{\frac{x}{1-\frac{x-1}{x}}}

and decide which of the answer choices it is equal to.

Pick an answer.

(A)

-x

(B)

(C)

(D)

$\sqrt{\frac{x}{2}}$

(E)

$x\sqrt{-1}$

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The expression is a fraction stacked inside a square root, so break it into small pieces: first clean up the little denominator, then do the division, then take the root. Because x is negative, the last piece needs care with the sign. A test value like x=-2 then checks the result and rules out the look-alike choices.

1STEP 1

Clean up the bottom

Write $1$ as $\frac{x}{x}$ and subtract the numerators, $x-(x-1)=1$ , so the whole bottom collapses to $\frac{1}{x}$ .

1-(x-1)/x=x/x-(x-1)/x=(x-(x-1))/x=1/x

2STEP 2

Do the division

Dividing by $\frac{1}{x}$ is the same as multiplying by its reciprocal $x$ , so under the root sits $x \cdot x$ , namely $x^2$ .

x/1/x=x · x=x²

3STEP 3

Take the square root

$\sqrt{x^2}$ is the distance of $x$ from zero, the absolute value $|x|$ , which is never negative.

√(x²)=|x|

4STEP 4

Use that x is negative

Since $x$ is negative, $|x|$ is its opposite $-x$ , which is positive; $x=-2$ checks out, giving $2$ both ways — choice (A).

x ＜ 0 → |x|=-x

Answer

-x

The result -x is positive when x ＜ 0, which is exactly what a real square root should give. Plugging in x=-2 makes the original expression equal 2 and -x equal 2, so they agree. Choice (B) x would be negative, and choice (E) x√(-1) is not even a real number, so neither can be the value of a real root.

💡Key takeaway

The square root of x² is the size of x, so when x is negative the answer is -x, which comes out positive.

Clean up the bottom
Do the division
Take the square root
Use that x is negative

Clean up the bottom

Do the division

Take the square root

Use that x is negative

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