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 Only one of the equations does not require Step 1 of the method for completing the square described in this section. Which one is it? Solve it.

Verified step by step guidance

1. Step 1: Identify the equation that does not require completing the square. Completing the square is typically used for quadratic equations in the form \(ax^2 + bx + c = 0\).
2. Step 2: Analyze each option: A. \(3x^2 - 17x - 6 = 0\) is a quadratic equation in standard form. B. \((2x + 5)^2 = 7\) is already in a form that can be solved by taking the square root. C. \(x^2 + x = 12\) is a quadratic equation in standard form. D. \((3x - 1)(x - 7) = 0\) is factored and can be solved using the zero product property.
3. Step 3: Recognize that option D, \((3x - 1)(x - 7) = 0\), does not require completing the square because it is already factored.
4. Step 4: Solve the equation \((3x - 1)(x - 7) = 0\) by setting each factor equal to zero: \(3x - 1 = 0\) and \(x - 7 = 0\).
5. Step 5: Solve each equation separately: For \(3x - 1 = 0\), add 1 to both sides and then divide by 3. For \(x - 7 = 0\), add 7 to both sides.

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming them into a perfect square trinomial. This involves rearranging the equation and adding a specific value to both sides to create a binomial squared. This technique is particularly useful for deriving the quadratic formula and analyzing the properties of quadratic functions.

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. The solutions to these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for identifying the appropriate solving method.

Factoring

Factoring is the process of breaking down an expression into a product of simpler expressions, which can make solving equations easier. For quadratic equations, this often involves finding two binomials that multiply to give the original quadratic. Recognizing when an equation can be factored directly can save time and simplify the solving process.