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In many quadratic equations, especially those involving square roots, the first step is isolating the square root term. This makes it simpler to deal with and allows the rest of the equation to be manipulated more easily.

When beginning with the equation \(2x - \sqrt{x} - 1 = 0\), your goal is to get \(\sqrt{x}\) by itself on one side of the equation. To do this, you add \(\sqrt{x} + 1\) to both sides of the equation, which gives you:

• \(2x - 1 = \sqrt{x}\)

This is an essential step, as it separates the square root so you can eliminate it through squaring. After isolation, you manage to transform the problem into one where you can apply algebraic manipulation like squaring both sides to further simplify and solve the equation. Remember, isolating square roots is crucial to correctly handling the problem and will set you on a path to finding possible solutions.

Once the square root has been isolated and the equation has been squared, you often end up with a quadratic equation, which is typical after squaring both sides. In this instance, the equation becomes \(4x^2 - 4x + 1 = x\). After rearranging—as shown by moving \(x\) to one side—you get the classic form:

• \(4x^2 - 5x + 1 = 0\)

This is now ready for the powerful quadratic formula:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For our quadratic equation, we have \(a = 4\), \(b = -5\), and \(c = 1\). By substituting these into the formula, we can find the potential solutions:

• \(x = \frac{5 \pm \sqrt{9}}{8}\), which simplifies to \(x = 1\) and \(x = \frac{1}{4}\)

Using the quadratic formula lets us solve for \(x\) efficiently, particularly when the quadratic cannot be easily factored. This handy formula is a go-to method for solving such equations.

After solving any quadratic equation, verifying your solutions is a necessary step to ensure accurate results. This is because not all solutions derived might satisfy the original equation, especially when initial equations involve square roots. For the solutions derived as \(x = 1\) and \(x = \frac{1}{4}\), you insert them back into the initial equation \(2x - \sqrt{x} - 1 = 0\):

• For \(x = 1\), the equation simplifies perfectly to \(2 - 1 - 1 = 0\), which is correct.
• For \(x = \frac{1}{4}\), it results in \(\frac{1}{2} - \frac{1}{2} - 1 eq 0\), clearly indicating that it does not satisfy the original equation.

Verifying is crucial because earlier steps like squaring both sides can introduce extraneous solutions. Confirming which solutions hold true maintains the integrity of the mathematical process and ensures understanding of the methodology.