"If (4 − x)/(2 + x) = x, What Is the Value of x² + 3x − 4?" — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

If $\dfrac{4 - x}{2 + x} = x$, what is the value of $x^2 + 3x - 4$?

(A) $-4$

(B) $-1$

(C) $0$

(D) $1$

(E) $2$

Try it before reading on.

Step 1: Rearrange the Given Equation

Start with the given equation:

$\dfrac{4 - x}{2 + x} = x$

Multiply both sides by $(2 + x)$ to clear the fraction:

$4 - x = x(2 + x)$

$4 - x = 2x + x^2$

Now move everything to one side to form a standard quadratic equation. Subtract $4 - x$ from both sides:

$0 = x^2 + 2x + x - 4$

$0 = x^2 + 3x - 4$

Step 2: Read What the Question Actually Asks

The trap lives in the difference between what the question asks and what the pattern suggests you should do.

The question asks: "What is the value of $x^2 + 3x - 4$?"

We just derived that $x^2 + 3x - 4 = 0$.

The answer is (C).

The question is asking for the value of the expression, and the equation we derived tells us directly that the value is $0$.

Why This Problem Matters

About 17% of test takers miss this problem, and the wrong answers cluster heavily on two choices: (D) $1$ at 9% and (A) $-4$ at 5%.

Both wrong answers come from doing correct math for the wrong question. Here's what happens:

You rearrange the equation and get $x^2 + 3x - 4 = 0$. You recognize this as a quadratic. Your muscle memory says "factor it and find the roots." You factor it correctly:

$x^2 + 3x - 4 = (x + 4)(x - 1) = 0$

Setting each factor to zero gives $x = -4$ and $x = 1$. And there they are in the answer choices: (A) is $-4$ and (D) is $1$.

Both roots are correct solutions to the quadratic equation. Both are wrong answers to the question on the page.

The question asks for the value of $x^2 + 3x - 4$, not for the values of $x$ that make $x^2 + 3x - 4$ equal zero. Those are different questions. The first one asks "what is this expression equal to?" The second asks "what makes this expression equal zero?" The rearrangement already answered the first question. The factoring answered a question nobody asked.

This is one of the most instructive problem types on the GMAT® because it separates process from knowledge. You can know algebra cold, execute every step flawlessly, and still pick a wrong answer because you solved for $x$ when the question asked for the value of an expression.

The fix is a re-read habit. After you finish computing, before you select an answer, re-read the question's exact wording. "What is the value of $x^2 + 3x - 4$?" is not the same question as "What are the solutions to $x^2 + 3x - 4 = 0$?" The words "value of" and "solutions to" are doing different work, and that difference is worth three seconds of your time to verify.

If you find yourself missing problems where you did all the math correctly, the issue is almost always process, not content. A flashcard won't fix it. A re-read habit will.

Want the full strategy behind this problem? Read: GMAT® Quant Fundamentals: Process Over Computation

From Episode 33 of Real GMAT® Problems (The GMAT® Strategy Podcast).

Ready for the next one? "3.003 ÷ 2.002 =..." — GMAT® Worked Solution.