Solve each equation in Exercises 15–34 by the square root property. (4x - 1)^2 = 16

Verified step by step guidance

1. Start by applying the square root property to both sides of the equation: \((4x - 1)^2 = 16\).
2. Take the square root of both sides to eliminate the square: \(4x - 1 = \pm \sqrt{16}\).
3. Simplify the square root on the right side: \(4x - 1 = \pm 4\).
4. Set up two separate equations to solve for \(x\): \(4x - 1 = 4\) and \(4x - 1 = -4\).
5. Solve each equation for \(x\) by isolating \(x\) on one side.

Key Concepts

Square Root Property

The square root property states that if a quadratic equation is in the form (ax + b)^2 = c, then the solutions can be found by taking the square root of both sides. This leads to two possible equations: ax + b = √c and ax + b = -√c. 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 first simplifying the equation to the form (4x - 1)^2 = 16, and then applying the square root property to solve for x. This step is crucial for finding the correct solutions.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. They can be solved using various methods, including factoring, completing the square, and using the quadratic formula. Understanding the nature of quadratic equations helps in recognizing when to apply the square root property effectively.