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Solving square root equations, such as the one given in the exercise \(2x + 1 = \sqrt{9x}\), frequently involves isolating the square root on one side of the equation and then squaring both sides to remove the square root. This is a sensitive operation because squaring both sides can introduce extraneous solutions—solutions that don't actually satisfy the original equation.

To prevent mistakes, it's important to clearly perform each step. After squaring, you get a quadratic equation that can be solved using standard methods. Once you've found the potential solutions, always substitute them back into the original equation to verify their validity, as some might be extraneous due to the squaring step.

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is an invaluable tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It allows you to find the solutions algebraically by substituting the coefficients \(a\), \(b\), and \(c\) from your equation into the formula.

Understanding the details is crucial. The term under the square root, \(b^2 - 4ac\), is called the discriminant. It can tell you about the nature of the roots even before you calculate them: if it's positive, expect two real solutions; if it's zero, one real solution; and if it's negative, the solutions are complex. Always apply the quadratic formula correctly, and simplify your radicals to get the most accurate results.

Simplifying algebraic expressions is a fundamental skill in solving equations. For the exercise \(4x^2 + 4x + 1 = 81x\), simplification involves combining like terms, leading you to the quadratic equation \(4x^2 - 5x + 1 = 0\).

When simplifying, adhere to the distributive property and combine like terms carefully to end with the simplest form possible. Simplification might not always look like it's making the expression simpler, but it's about preparing the equation for the next steps of solving it, which could include factorisation or application of formulas, as in this exercise where we need a simplified quadratic equation to apply the quadratic formula efficiently.