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When we talk about solving equations, especially quadratic ones, we're referring to finding the values of the variable that make the equation true. For example, in the equation \(x^2 - \sqrt{5} = 0\), our goal is to determine which values of \(x\) satisfy this equation.

Solving equations often involves several steps including manipulation of the equation to isolate the variable. In our particular equation, this process involves getting \(x\) by itself on one side of the equation.

The key steps generally include:

• Rearranging terms to simplify the equation
• Applying mathematical operations like addition, subtraction, multiplication, or division to both sides to get the variable alone
• Checking the solution by plugging it back into the original equation

These methods allow us to find the value(s) that \(x\) can take to keep the equation balanced and true.

A square root is a special mathematical function that essentially "reverses" the process of squaring a number. If you square a number \(x\), you are multiplying it by itself (\(x^2\)). The square root, denoted as \(\sqrt{}\), asks the question: "What number times itself gives me this number?"

In our equation \(x^2 = \sqrt{5}\), we solve for \(x\) by applying the square root function to both sides. It's important to note that taking the square root of both sides of an equation means considering both the positive and negative roots. Thus, we get two potential solutions: \(x = \sqrt[4]{5}\) and \(x = -\sqrt[4]{5}\).

This concept is crucial because squaring and taking square roots are inverse operations—a fact that's frequently utilized in solving equations.

Algebraic manipulation involves applying various operations to rewrite and simplify equations, making them easier to solve. In our example equation \(x^2 - \sqrt{5} = 0\), the manipulation includes repositioning terms and managing operations to reach a solution.

The steps in the original solution involve adding \(\sqrt{5}\) to both sides to remove it from the left-hand side. This gives us \(x^2 = \sqrt{5}\). Such manipulation is common in algebra, as it allows the expression to be transformed without changing its equality.

Key techniques in algebraic manipulation include:

• Addition or subtraction of the same value to both sides
• Multiplication or division by non-zero numbers
• Substituting equivalent expressions

Mastery of these techniques is fundamental for solving a wide range of mathematical problems.