Verified Answer :

  • Determine the domain restriction \(x \geq 6\) by ensuring both sides of the equation are non-negative.
  • Transform the equation into the quadratic form \(x^2 - 14x + 40 = 0\) by isolating the square root and squaring both sides.
  • Factor the quadratic equation to find potential candidates \(x = 10\) and \(x = 4\).
  • Validate the candidates against the domain restriction to conclude that the only valid solution is \(\boxed{x=10}\).

Explanation

  1. Analysis and Domain Restriction

To find the solution to the equation \(\sqrt{2x - 4} - x + 6 = 0\), we first identify the domain of the variable \(x\). The term \(\sqrt{2x - 4}\) is defined only when the expression inside the square root is non-negative, which means \(2x - 4 \geq 0\), leading to \(x \geq 2\). Furthermore, we can rewrite the original equation as \(\sqrt{2x - 4} = x - 6\). Since the square root function outputs non-negative values, the right-hand side, \(x - 6\), must also be non-negative, requiring \(x - 6 \geq 0\), or \(x \geq 6\). Thus, any valid solution must satisfy \(x \geq 6\).

  1. Formulating and Solving the Quadratic Equation

With the domain established, we square both sides of the equation \(\sqrt{2x - 4} = x - 6\) to eliminate the square root:
\[ (\sqrt{2x - 4})^2 = (x - 6)^2 \]
This simplifies to:
\[ 2x - 4 = x^2 - 12x + 36 \]
Rearranging this into a standard quadratic form \(ax^2 + bx + c = 0\) by moving all terms to one side, we get:
\[ x^2 - 14x + 40 = 0 \]

  1. Verification of Roots and Final Conclusion

To solve the quadratic equation \(x^2 - 14x + 40 = 0\), we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) or factor the trinomial. Factoring gives:
\[ (x - 10)(x - 4) = 0 \]
This yields two potential solutions: \(x = 10\) and \(x = 4\). We must now verify these against our initial domain restriction of \(x \geq 6\). Checking \(x = 4\), we see that \(4 < 6\), so this is an extraneous solution introduced by squaring. Checking \(x = 10\), we find it satisfies \(x \geq 6\). Verification in the original equation: \(\sqrt{2(10) - 4} - 10 + 6 = \sqrt{16} - 4 = 4 - 4 = 0\). Thus, \(x = 10\) is the only valid solution.

Examples

Imagine you are designing a ramp for a skateboard park where the height of the platform needs to change based on the ramp's length, and this height follows a square root relationship. If the formula relating the height of the ramp to its distance from a fixed point is \(\sqrt{2x - 4} = x - 6\), calculating the value of \(x\) correctly ensures the ramp is stable and safe. Solving such equations is crucial for engineering and construction tasks to ensure that all structural components fit together perfectly and function as intended.

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