Verified Answer :

  • Identify the domain condition \(x \geq 1.5\) to ensure the equality holds.
  • Square both sides of the equation and simplify to isolate the radical: \(4\sqrt{x+9} = x - 12\).
  • Square both sides again to obtain the quadratic equation \(x^2 - 40x = 0\).
  • Identify the potential solutions \(x = 0\) and \(x = 40\), and verify them in the original equation, finding that only \(x = 40\) is valid.
  • The final answer is \(\boxed{40}\).

Explanation

  1. Domain Analysis

The given equation is \(\sqrt{x+9} + 2 = \sqrt{2x+1}\). First, we identify the domain of the equation. The expressions inside the square roots must be non-negative:

  1. \(x + 9 \geq 0 \implies x \geq -9\)
  2. \(2x + 1 \geq 0 \implies x \geq -1/2\)

Combining these, we must have \(x \geq -1/2\). Additionally, since \(\sqrt{x+9} \geq 0\), the left side is at least 2, so \(\sqrt{2x+1}\) must be at least 2, which means \(2x + 1 \geq 4 \implies 2x \geq 3 \implies x \geq 1.5\). This is a more restrictive condition than \(x \geq -1/2\).

  1. Squaring and Simplifying the Equation

To eliminate the square root, we square both sides of the equation:
\[(\sqrt{x+9} + 2)^2 = (\sqrt{2x+1})^2\]
Expanding the left side using \((a+b)^2 = a^2 + 2ab + b^2\) with \(a = \sqrt{x+9}\) and \(b = 2\):
\[(x+9) + 4\sqrt{x+9} + 4 = 2x+1\]
\[x + 13 + 4\sqrt{x+9} = 2x + 1\]
Isolating the square root term:
\[4\sqrt{x+9} = 2x + 1 - x - 13\]
\[4\sqrt{x+9} = x - 12\]

  1. Solving the Quadratic Equation

We square both sides again to eliminate the remaining square root:
\[(4\sqrt{x+9})^2 = (x - 12)^2\]
\[16(x+9) = x^2 - 24x + 144\]
\[16x + 144 = x^2 - 24x + 144\]
Subtracting 144 from both sides and rearranging:
\[x^2 - 40x = 0\]
\[x(x - 40) = 0\]
This gives us two potential solutions: \(x = 0\) and \(x = 40\).

  1. Verification and Final Conclusion

Now we must check our potential solutions in the original equation \(\sqrt{x+9} + 2 = \sqrt{2x+1}\):

  1. For \(x = 0\): \(\sqrt{0+9} + 2 = 3 + 2 = 5\), while \(\sqrt{2(0)+1} = \sqrt{1} = 1\). Since \(5 eq 1\), \(x=0\) is an extraneous solution.
  2. For \(x = 40\): \(\sqrt{40+9} + 2 = \sqrt{49} + 2 = 7 + 2 = 9\), while \(\sqrt{2(40)+1} = \sqrt{81} = 9\). Since \(9 = 9\), \(x=40\) is a valid solution.

Thus, the only valid solution is \(\boxed{40}\).

Examples

Solving equations with radicals is essential in physics and engineering, such as determining the distance of an object moving under constant acceleration or calculating forces in mechanical systems. When a formula involves a square root that bounds the possible range of variables, solving for the correct value ensures that the physical conditions are satisfied, such as ensuring a velocity or distance result is positive and realistic.

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