Solve each equation in Exercises 15–34 by the square root property. 3x^2 - 1 = 47

Verified step by step guidance

1. Start by isolating the term with the square. Add 1 to both sides of the equation: \(3x^2 - 1 + 1 = 47 + 1\).
2. Simplify the equation: \(3x^2 = 48\).
3. Divide both sides by 3 to solve for \(x^2\): \(x^2 = \frac{48}{3}\).
4. Simplify the right side: \(x^2 = 16\).
5. Apply the square root property by taking the square root of both sides: \(x = \pm \sqrt{16}\).

Key Concepts

Square Root Property

The square root property states that if an equation is in the form of x^2 = k, then the solutions for x can be found by taking the square root of k. This property allows us to isolate the variable by applying the square root to both sides of the equation, leading to two possible solutions: x = ±√k.

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. This is a crucial step in solving equations, as it simplifies the problem and allows for the application of properties like the square root property to find the solution.

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. In this context, the equation 3x^2 - 1 = 47 can be transformed into a standard quadratic form, allowing the use of the square root property to solve for x.