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The square root property is an essential tool for solving quadratic equations, especially when the equation is in the form given by \((ax + b)^2 = c\). This property states that if \(u^2 = v\), then \(u = \pmot \sqrt{v}\). When applied to an equation: \((2x - 5)^2 = 180\), you can take the square root of both sides to simplify the equation. Remember,'t it is crucial to consider both the positive and negative roots when applying the square root property, resulting in: \(2x - 5 = \pm \sqrt{180}\). This will give you two potential solutions for \(2x - 5\).\.
Taking the square root helps simplify the complexity of the equation and paves the way for solving for \(x\), making it easier to isolate the variable in subsequent steps.

Simplifying radicals appears often when solving quadratic equations. When we encounter a term like \(\sqrt{180}\), we need to break it down into its prime factors for simplification. Let's look at this step-by-step process:

• First, write \(180\) as the product of its prime factors: \(180 = 36 \times 5\).
• Next, apply the square root to both factors: \(\sqrt{180} = \sqrt{36 \times 5}\).
• Then, simplify the square root of 36: \(\sqrt{36} = 6\), so the expression becomes: \(\sqrt{180} = 6\sqrt{5}\).

This process transforms a complex radical into a simpler form that you can then use in the equation. Simplifying radicals is key to making the equation much more manageable and finding the solution with accuracy.

Isolating the variable is the final step in solving for \(x\). Here, we use basic algebraic manipulations. After applying the square root property and simplifying the radical, we have: \(2x - 5 = \pm 6\sqrt{5}\). To isolate \(x\), follow these steps:

• First, add 5 to both sides of the equation: \(2x = 5 \pm 6\sqrt{5}\).
• Next, divide both sides by 2 to solve for \(x\): \(x = \frac{5 \pm 6\sqrt{5}}{2}\).

By following this method, you can solve for the exact value of \(x\). Ensuring to perform each step carefully and keeping the operations balanced on both sides of the equation is crucial for arriving at the correct solution. Moreover, isolating the variable involves breaking down the steps logically and consistently, making the entire process simpler and more effective.