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 Only one of the equations is set up so that the values of a, b, and c can be determined immediately. Which one is it? Solve it.

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Identify the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.

Examine each given equation to see which one is already written in this standard form:

A. \(3x^2 - 17x - 6 = 0\) is in the form \(ax^2 + bx + c = 0\) with \(a=3\), \(b=-17\), and \(c=-6\).

B. \((2x + 5)^2 = 7\) is not in standard form; it needs to be expanded and rearranged.

C. \(x^2 + x = 12\) needs to be rearranged by subtracting 12 from both sides to get \(x^2 + x - 12 = 0\).

D. \((3x - 1)(x - 7) = 0\) is factored form; it can be expanded to standard form but is not immediately in that form.

Since equation A is already in standard form, use the quadratic formula to solve it: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Substitute \(a=3\), \(b=-17\), and \(c=-6\) into the quadratic formula and simplify under the square root and the entire expression to find the solutions for \(x\).

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주요 개념

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Standard Form of a Quadratic Equation

A quadratic equation is in standard form when written as ax² + bx + c = 0, where a, b, and c are constants. This form allows immediate identification of coefficients needed for solving the equation using methods like the quadratic formula.

Identifying Coefficients a, b, and c

To solve a quadratic equation using formulas, you must know the values of a, b, and c. These are the coefficients of x², x, and the constant term, respectively, and must be clearly visible or easily extracted from the equation.

Solving Quadratic Equations by Factoring

Factoring involves expressing a quadratic as a product of binomials set equal to zero. This method is efficient when the equation is factorable, allowing you to find solutions by setting each factor equal to zero and solving for x.

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 Only one of the equations is set up so that the values of a, b, and c can be determined immediately. Which one is it? Solve it.