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

Verified step by step guidance

1. Start by applying the square root property to both sides of the equation. The square root property states that if \((a)^2 = b\), then \(a = \pm \sqrt{b}\). For this problem, take the square root of both sides: \(\sqrt{(x + 2)^2} = \pm \sqrt{25}\).
2. Simplify both sides of the equation. The square root of \((x + 2)^2\) is \(x + 2\), and the square root of 25 is 5. This gives \(x + 2 = \pm 5\).
3. Split the equation into two separate cases to account for the \(\pm\) symbol. Case 1: \(x + 2 = 5\). Case 2: \(x + 2 = -5\).
4. Solve each case for \(x\). For Case 1: Subtract 2 from both sides to get \(x = 5 - 2\). For Case 2: Subtract 2 from both sides to get \(x = -5 - 2\).
5. Write the solutions as a set. The solutions are the values of \(x\) obtained from both cases.

Key Concepts

Square Root Property

The square root property states that if x^2 = a, then x = ±√a. This property allows us to solve quadratic equations by taking the square root of both sides, leading to two possible solutions: one positive and one negative. It is essential for solving equations where the variable is squared.

Isolating the Variable

Isolating the variable involves rearranging an equation to get the variable on one side and all other terms on the opposite side. In the context of the square root property, this often means first simplifying the equation to the form x^2 = a before applying the square root. This step is crucial for correctly applying the property.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique is particularly useful when the equation is not in standard form. It helps in deriving the square root property and can also be used to derive the quadratic formula.