Verified Answer :

  • Define the domain of the equation: \(x \geq -1\).
  • Observe that the left-hand side \(2 \sqrt{x+1}\) is always non-negative (\(\geq 0\)).
  • Observe that the right-hand side \(-\sqrt{x+2}\) is always negative (\(\leq -1\)).
  • Conclude that there is no real solution as the sides cannot be equal.
  • Final answer: \(\boxed{\text{no solution}}\).

Explanation

  1. Analyzing the Domain

The equation to solve is \(2 \sqrt{x+1} = -\sqrt{x+2}\). To find the real-valued solution, we must first consider the domain of the expression. For the square root terms to be defined in the set of real numbers, the expressions under the radicals must be non-negative:

  • \(x + 1 \geq 0 \implies x \geq -1\)
  • \(x + 2 \geq 0 \implies x \geq -2\)

Combining these requirements, the domain of the equation is \(x \geq -1\).

  1. Comparing the Ranges of the Sides

Next, we analyze the range of both sides of the equation within the domain \(x \geq -1\).

  • The left-hand side is \(L(x) = 2 \sqrt{x+1}\). Since the square root function \(\sqrt{u}\) returns a non-negative value for any non-negative \(u\), \(L(x) \geq 0\) for all \(x \geq -1\).
  • The right-hand side is \(R(x) = -\sqrt{x+2}\). Since \(\sqrt{x+2} \geq 0\) for all \(x \geq -1\), the value of \(R(x) = -\sqrt{x+2}\) must be less than or equal to \(0\).

Specifically, \(R(x) = -\sqrt{x+2} \leq -\sqrt{-1+2} = -1\) for all \(x \geq -1\).

  1. Conclusion

We have determined that for any \(x\) in the domain \(x \geq -1\):

  • The left-hand side \(L(x)\) is always non-negative (\(L(x) \geq 0\)).
  • The right-hand side \(R(x)\) is always negative (\(R(x) \leq -1\)).

Since a non-negative value can never equal a negative value, there is no value of \(x\) that satisfies the given equation. Thus, there is no real solution.

Examples

In real-life scenarios, square root equations often represent physical constraints where values must remain non-negative. For example, if you are measuring the length or speed of an object represented by a function of time \(t\), these values must be greater than or equal to zero. If your mathematical model results in an equation where a non-negative measurement is equated to a negative value (e.g., a speed equal to a negative distance), it indicates that the conditions described by your model are impossible or that no such physical state exists. This is a common way to identify impossible physical situations in engineering and physics.

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