CONCEPT PREVIEW Use choices A–D to answer each question.


A. 3x² - 17x - 6 = 0
B.(2x + 5)² = 7
C. x² + x = 12
D. (3x - 1) (x - 7) = 0


Which quadratic equation is set up for direct use of the square root property? Solve it.

Verified step by step guidance

1. Identify the quadratic equation that is set up for direct use of the square root property. The square root property is applicable when the equation is in the form \((ax + b)^2 = c\).
2. Examine each option: A. \(3x^2 - 17x - 6 = 0\), B. \((2x + 5)^2 = 7\), C. \(x^2 + x = 12\), D. \((3x - 1)(x - 7) = 0\).
3. Notice that option B, \((2x + 5)^2 = 7\), is already in the form \((ax + b)^2 = c\), which is suitable for applying the square root property.
4. Apply the square root property to option B: Take the square root of both sides to get \(2x + 5 = \pm \sqrt{7}\).
5. Solve for \(x\) by isolating it: Subtract 5 from both sides and then divide by 2 to find the values of \(x\).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + 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, and applying the quadratic formula. Understanding the structure of quadratic equations is essential for identifying which method to use for solving them.

Square Root Property

The square root property states that if x² = k, then x = ±√k. This property is particularly useful for solving quadratic equations that can be expressed in the form (x - p)² = k, allowing for a straightforward solution by taking the square root of both sides. Recognizing when to apply this property is crucial for efficiently solving certain quadratic equations.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic equation as a product of its linear factors. This method is effective when the equation can be expressed in the form (ax + b)(cx + d) = 0. Understanding how to factor quadratics is important for identifying solutions quickly and is often a prerequisite for applying the square root property.