Use Choices A–D to answer each question. A. 3x^2 - 17x - 6 = 0 B. (2x + 5)^2 = 7 C. x^2 + x = 12 D. (3x - 1)(x - 7) = 0 Which equation is set up for direct use of the square root property? Solve it

Verified step by step guidance

1. Identify the equation that is set up for direct use of the square root property. The square root property is typically used when an equation is in the form \((ax + b)^2 = c\).
2. Look at the given choices: A. \(3x^2 - 17x - 6 = 0\), B. \((2x + 5)^2 = 7\), C. \(x^2 + x = 12\), D. \((3x - 1)(x - 7) = 0\).
3. Choice B, \((2x + 5)^2 = 7\), is in the form \((ax + b)^2 = c\), which is suitable for applying the square root property.
4. To solve \((2x + 5)^2 = 7\) using the square root property, take the square root of both sides: \(2x + 5 = \pm \sqrt{7}\).
5. Solve for \(x\) by isolating it: Subtract 5 from both sides to get \(2x = -5 \pm \sqrt{7}\), then divide by 2 to find \(x\).

Key Concepts

Square Root Property

The Square Root Property states that if an equation is in the form x^2 = k, where k is a non-negative number, then the solutions for x can be found by taking the square root of k. This property allows us to solve quadratic equations directly by isolating the squared term and applying the square root to both sides, yielding two possible solutions: x = ±√k.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations can be solved using various methods, including factoring, completing the square, or applying the quadratic formula. Recognizing the standard form is crucial for determining the appropriate method for solving the equation.

Factoring

Factoring involves rewriting an expression as a product of its factors, which can simplify solving equations. For quadratic equations, this often means expressing the equation in the form (px + q)(rx + s) = 0. This method is particularly useful when the equation can be easily decomposed into simpler binomials, allowing for the application of the Zero Product Property to find the roots.