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Understanding how to manipulate square roots is crucial in solving equations involving radicals. A square root symbol, \( \sqrt{} \), essentially asks for a number that, when multiplied by itself, will yield the original number inside the root. An important property to remember is that \( \sqrt{a^2} = |a| \), which acknowledges the absolute value since both positive and negative numbers squared will give a positive result.

When faced with an equation like \( \sqrt{x^2 + 3} = \sqrt{28} \), the first instinct should be to eliminate the square roots to simplify the problem. This is done by squaring both sides, taking care to apply the exponent to the entire contents of the square root. After this step, you are often left with a standard algebraic equation to solve. However, one must be mindful during the step of squaring as it might introduce extraneous solutions. That's why checking the solutions later is an essential part of the process.

Isolating the variable is a foundational step in solving almost any algebraic equation. The goal is to have the variable on one side of the equation and all other terms on the opposite side. This process often involves basic arithmetic operations: addition or subtraction to move terms across the equal sign and multiplication or division to consolidate or disperse coefficient values.

For instance, for the squared form of our equation \(x^2 + 3 = 28\), you would subtract 3 from both sides to get \(x^2 = 25\). Reaching this point sets you up perfectly to then perform the square root operation to find the value(s) of \(x\). It's vital to proceed with these steps logically and methodically, checking for any mistakes at each stage. An algebraic equation is a balance; what you do to one side, you do to the other to maintain equality.

After finding potential solutions, especially in radical equations, it is imperative to check each one to ensure that they actually satisfy the original equation. Some operations, like squaring both sides of an equation, can introduce solutions that don't hold true when plugged back in. This process of validation is known as 'checking your solutions.'

To check a solution, you substitute the solution back into the original equation and see if the equation 'balances' or makes sense. In our example, after determining that \( x = \pm 5 \), we replaced \( x \) with 5 and -5 in the original equation and confirmed that both solved the equation accurately. This final step is not just a formality; it actually confirms that the operations performed did not disturb the integrity of the original equation.