Verified Answer :

  • Define the domain: \(-4 \leq x \leq -2.5\).
  • Isolate and square the radical to get the quadratic equation \(x^2 + 10x + 21 = 0\).
  • Solve by factoring: \((x + 7)(x + 3) = 0\) gives potential roots \(-7\) and \(-3\).
  • Verify roots against the domain to find the only valid solution is \(-3\).
  • Final answer: \(\boxed{-3}\).

Explanation

  1. Analyzing the Domain of the Equation

We are given the equation \(\sqrt{-2x - 5} - 4 = x\). For a solution to exist, the expression inside the square root must be non-negative: \(-2x - 5 \geq 0\), which simplifies to \(-2x \geq 5\) or \(x \leq -\frac{5}{2}\). Furthermore, since the principal square root is always non-negative, \(\sqrt{-2x - 5} = x + 4\) implies that \(x + 4 \geq 0\), so \(x \geq -4\). Thus, any valid solution must satisfy \(-4 \leq x \leq -\frac{5}{2}\).

  1. Isolating and Squaring the Radical

To solve the equation, we first isolate the radical:
\[\sqrt{-2x - 5} = x + 4\]
Squaring both sides gives:
\[-2x - 5 = (x + 4)^2\]
Expanding the right side:
\[-2x - 5 = x^2 + 8x + 16\]
Rearranging all terms to one side:
\[x^2 + 10x + 21 = 0\]

  1. Solving the Quadratic Equation

We now solve the quadratic equation \(x^2 + 10x + 21 = 0\) by factoring. We seek two numbers that multiply to 21 and add to 10. These numbers are 7 and 3. Thus, the equation can be written as:
\[(x + 7)(x + 3) = 0\]
Setting each factor to zero, we get two potential solutions:
\(x = -7\) and \(x = -3\).

  1. Verifying Potential Solutions

We must check these values against our original domain condition, \(-4 \leq x \leq -2.5\).

  • For \(x = -7\): The value \(-7\) is less than \(-4\), which is outside the required interval. Substituting back into the original equation, \(\sqrt{-2(-7) - 5} - 4 = \sqrt{14-5} - 4 = \sqrt{9} - 4 = 3 - 4 = -1\), which does not equal \(-7\). Thus, \(x = -7\) is an extraneous solution.
  • For \(x = -3\): The value \(-3\) is within the interval \([-4, -2.5]\). Substituting back into the original equation, \(\sqrt{-2(-3) - 5} - 4 = \sqrt{6 - 5} - 4 = \sqrt{1} - 4 = 1 - 4 = -3\), which equals \(x\). Thus, \(x = -3\) is the valid solution.
  1. Conclusion

Based on our analysis, the only solution to the equation \(\sqrt{-2x - 5} - 4 = x\) is \(x = -3\). Comparing this to the given choices, the correct option is \(-3\). Final answer is \(\boxed{-3}\).

Examples

Equations involving radicals appear frequently in engineering and physics, such as calculating the distance to an object or analyzing wave propagation. For instance, when designing a sensor system, you might have an equation where the response time is related to the square root of some input variables. Identifying extraneous solutions is a critical skill to ensure that the calculated parameters are physically meaningful and accurate for real-world applications.

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