Solve each equation in Exercises 15–34 by the square root property. (8x - 3)^2 = 5

Verified step by step guidance

1. Start by recognizing that the equation is in the form \((ax + b)^2 = c\), which is suitable for applying the square root property.
2. Apply the square root property: If \(u^2 = c\), then \(u = \pm \sqrt{c}\). Here, let \(u = 8x - 3\), so \((8x - 3)^2 = 5\) becomes \(8x - 3 = \pm \sqrt{5}\).
3. Solve for \(x\) by first isolating \(8x\). Add 3 to both sides of the equation: \(8x = 3 \pm \sqrt{5}\).
4. Divide each term by 8 to solve for \(x\): \(x = \frac{3 \pm \sqrt{5}}{8}\).
5. This gives the two possible solutions for \(x\): \(x = \frac{3 + \sqrt{5}}{8}\) and \(x = \frac{3 - \sqrt{5}}{8}\).

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 results in 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 the equation to get the variable on one side and the constants on the other. In the context of the square root property, this often means first simplifying the equation to the form (ax + b)^2 = c before applying the square root. This step is crucial for accurately solving for the variable.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. They can be solved using various methods, including factoring, completing the square, and using the quadratic formula. Understanding the nature of quadratic equations is vital for applying the square root property effectively.