Solve each equation in Exercises 15–34 by the square root property.(x + 2)^2 = 25

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Start by recognizing that the equation \((x + 2)^2 = 25\) is in the form \(a^2 = b\), where \(a = x + 2\) and \(b = 25\).

Apply the square root property, which states that if \(a^2 = b\), then \(a = \pm \sqrt{b}\).

Take the square root of both sides of the equation: \(x + 2 = \pm \sqrt{25}\).

Simplify the square root: \(\sqrt{25} = 5\), so the equation becomes \(x + 2 = \pm 5\).

Solve for \(x\) by setting up two separate equations: \(x + 2 = 5\) and \(x + 2 = -5\), then solve each for \(x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Square Root Property

The square root property states that if a quadratic equation is in the form (x - a)² = b, then the solutions for x can be found by taking the square root of both sides. This results in two possible equations: x - a = √b and x - a = -√b. This property is essential for solving equations that can be expressed as perfect squares.

Isolating the Variable

Isolating the variable involves rearranging an equation to get the variable on one side and the constants on the other. In the context of the square root property, this means ensuring that the squared term is alone on one side of the equation before applying the square root. This step is crucial for correctly applying the square root property.

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. In the equation (x + 2)² = 25, recognizing that 25 is a perfect square (5²) allows us to apply the square root property effectively. Understanding perfect squares helps in identifying potential solutions and simplifying the solving process.

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