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Algebraic equations are mathematical statements expressing the equality between two algebraic expressions that consist of variables, numbers, and arithmetic operations. When dealing with algebraic equations, one essentially seeks to find the value of the variable that makes the equation true. For example, in the equation \(2x - 3 = 7\), we determine that when \(x\) is equal to 5, the equation holds true because \(2(5) - 3 = 7\).

To solve algebraic equations, specific strategies are applied based on equation properties. Simple equations can be solved by basic operations like addition, subtraction, multiplication, and division to isolate the variable. More complex ones, such as those involving radical expressions, require knowledge of higher-level algebraic concepts, including how to manipulate roots and powers.

A square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). Notated as \(\sqrt{x}\), it's the inverse operation of squaring a number. For instance, \(\sqrt{9} = 3\) because \(3^2 = 9\).

Properties of Square Roots

Square roots have unique properties that are crucial when solving equations involving them. It's important to remember that every positive number has two square roots: a positive and a negative one. However, when we see \(\sqrt{x}\), it refers to the principal (or positive) square root. To handle equations with square roots, we often square both sides to eliminate the root, simplifying the equation, as seen in the given exercise where \(\sqrt{2x - 5} = \sqrt{x + 4}\) becomes \(2x - 5 = x + 4\) after squaring both sides.

Extraneous solutions are results that emerge during the process of solving an equation but are not valid solutions to the original equation. They often appear when performing operations that are not reversible, such as squaring both sides of an equation containing a square root. Squaring can create solutions that weren't part of the solution set of the original equation.

This is why any potential solution must be substituted back into the original equation to verify its validity. In our exercise, after isolating \(x\) and finding that \(x = 9\), we check it against the original equation to ensure it isn't an extraneous solution. The verification process confirmed that \(x = 9\) does not produce contradictions when plugged back in, hence, being the true solution to the equation.