Verified Answer :

  • Identify the domain of the equation as \(x \geq 2\).
  • Square both sides, rearrange the terms, and isolate the remaining square root to form the quadratic equation \(x^2 - 30x + 81 = 0\).
  • Use the quadratic formula to find candidate roots \(x = 3\) and \(x = 27\).
  • Test each root in the original equation to find that only \(x = 27\) is valid, while \(x = 3\) is an extraneous solution.
  • The final solution is \(\boxed{27}\).

Explanation

  1. Problem Analysis and Domain Constraints

The given equation is \(\sqrt{3x} + 1 = \sqrt{4x - 8}\). Before solving, we must identify the domain of the expression, as square roots are only defined for non-negative values.

  1. The term \(\sqrt{3x}\) requires \(3x \geq 0\), which simplifies to \(x \geq 0\).
  2. The term \(\sqrt{4x - 8}\) requires \(4x - 8 \geq 0\), which simplifies to \(4x \geq 8\), or \(x \geq 2\).

Combining these conditions, the domain for the equation is \(x \geq 2\). Additionally, note that for the equation to hold, the right-hand side \(\sqrt{4x - 8}\) must be greater than or equal to 1, since the left-hand side is \(\sqrt{3x} + 1\). This implies \(\sqrt{4x - 8} \geq 1\), so \(4x - 8 \geq 1\), which means \(x \geq \frac{9}{4}\). Thus, we are looking for solutions such that \(x \geq 2.25\).

  1. Squaring Both Sides

To solve the equation \(\sqrt{3x} + 1 = \sqrt{4x - 8}\), we square both sides to eliminate the radicals:
\[(\sqrt{3x} + 1)^2 = (\sqrt{4x - 8})^2\]
Expanding the left side:
\[3x + 2\sqrt{3x} + 1 = 4x - 8\]
Next, isolate the radical term:
\[2\sqrt{3x} = 4x - 3x - 8 - 1\]
\[2\sqrt{3x} = x - 9\]

  1. Forming the Quadratic Equation

To eliminate the remaining radical, square both sides again:
\[(2\sqrt{3x})^2 = (x - 9)^2\]
\[4(3x) = x^2 - 18x + 81\]
\[12x = x^2 - 18x + 81\]
Rearranging into a standard quadratic equation form \(ax^2 + bx + c = 0\):
\[x^2 - 30x + 81 = 0\]

  1. Solving the Quadratic Equation

We solve the quadratic equation \(x^2 - 30x + 81 = 0\) using the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
\[x = \frac{30 \pm \sqrt{(-30)^2 - 4(1)(81)}}{2(1)}\]
\[x = \frac{30 \pm \sqrt{900 - 324}}{2} = \frac{30 \pm \sqrt{576}}{2} = \frac{30 \pm 24}{2}\]
This gives us two candidate solutions:

  1. \(x = \frac{30 + 24}{2} = \frac{54}{2} = 27\)
  2. \(x = \frac{30 - 24}{2} = \frac{6}{2} = 3\)
  1. Verification and Conclusion

Finally, we must verify the candidate solutions against the original equation and our domain constraints.

  • For \(x = 27\): The left side is \(\sqrt{3(27)} + 1 = \sqrt{81} + 1 = 9 + 1 = 10\). The right side is \(\sqrt{4(27) - 8} = \sqrt{108 - 8} = \sqrt{100} = 10\). Since \(10 = 10\), \(x = 27\) is a valid solution.
  • For \(x = 3\): The left side is \(\sqrt{3(3)} + 1 = \sqrt{9} + 1 = 3 + 1 = 4\). The right side is \(\sqrt{4(3) - 8} = \sqrt{12 - 8} = \sqrt{4} = 2\). Since \(4 eq 2\), \(x = 3\) is an extraneous solution.

Therefore, the only real solution is \(x = 27\).

Examples

Solving radical equations is a fundamental skill in modeling scenarios where variables are constrained by non-negative physical properties, such as measuring distances or physical growth rates. For example, if a company models its production growth using a radical function based on time and resource allocation, finding when two different growth models intersect (like the two sides of our equation) helps in making strategic timing decisions for scaling operations. By correctly identifying and discarding extraneous solutions, analysts ensure that their predictions remain grounded in reality, avoiding errors that could lead to inefficient resource management.

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