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The square root method is a useful technique for solving quadratic equations where the variable is isolated within a squared term. In the equation \((x+3)^{2}=16\), the goal is to eliminate the squared term by applying the square root to both sides. This step simplifies the problem because taking the square root and squaring are inverse operations that cancel each other out. Applying this method to the original equation, we take the square root of both sides, leading to \(x+3 = \pm \sqrt{16}\). This can be understood as finding two values for \(x+3\) that satisfy the equation, which is why we have a plus-minus (\(\pm\)) symbol. It's important to remember that both positive and negative roots must be considered, as squaring either a positive or negative number yields the same result. This is what helps find both possible solutions for \(x\).

Algebraic solutions are methods of finding the roots of an equation using algebraic manipulations. Once the square root method is applied to \((x+3)^{2}=16\), we get \(x+3 = \pm4\). To solve for \(x\), it involves applying simple algebraic steps. We solve each resulting equation separately:

• \(x+3 = 4\)
• \(x+3 = -4\)

For the first equation, subtracting 3 from both sides gives \(x = 4 - 3\), resulting in \(x = 1\). Similarly, for the second equation, subtracting 3 gives \(x = -4 - 3\), resulting in \(x = -7\). This provides two possible solutions or roots for the equation. Algebraic solutions often involve simple operations like addition, subtraction, and sometimes multiplication or division as we isolate the variable.

Equation simplification refers to the process of making an equation easier to work with. This usually involves reducing complex expressions to simpler forms. In the solution of \((x+3)^{2}=16\), after taking the square root of both sides, we simplify \(\sqrt{16}\). Knowing that the square root of 16 is 4 allows us to rewrite the equation as \(x+3 = \pm4\). This crucial simplification transforms the original quadratic expression into linear equations that are much more manageable to solve. A simplified form highlights the underlying structure of a mathematical problem, making it more straightforward to address. The key is recognizing factors and operations that can be combined or reduced, which is essential in solving algebraic equations effectively.