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 equation is set up for direct use of the zero-factor property? Solve it

Verified step by step guidance

1. Identify the equation that is already factored or can be easily factored to apply the zero-factor property. The zero-factor property states that if a product of factors equals zero, then at least one of the factors must be zero.
2. Look at each choice: Choice D is given as \((3x - 1)(x - 7) = 0\), which is already factored and set equal to zero, making it ready for direct use of the zero-factor property.
3. Apply the zero-factor property by setting each factor equal to zero separately: \$3x - 1 = 0\( and \)x - 7 = 0$.
4. Solve each equation for \(x\): For \$3x - 1 = 0\(, add 1 to both sides and then divide by 3. For \)x - 7 = 0$, add 7 to both sides.
5. Write the solutions as the values of \(x\) found from the two equations, which are the roots of the original equation.

Key Concepts

Zero-Factor Property

The zero-factor property states that if the product of two factors equals zero, then at least one of the factors must be zero. This property is used to solve equations set as a product of factors equal to zero by setting each factor individually to zero.

Factoring Quadratic Equations

Factoring involves rewriting a quadratic equation as a product of two binomials. This process is essential for applying the zero-factor property, as it transforms the equation into a form where each factor can be set to zero to find the solutions.

Solving Quadratic Equations

Solving quadratic equations can be done by factoring, completing the square, or using the quadratic formula. When an equation is factored and set equal to zero, the zero-factor property allows for straightforward solution by setting each factor equal to zero.