Solve each equation using the zero-factor property. See Example 1. 2x^2 - x = 15

Verified step by step guidance

1. Rewrite the equation in standard form by moving all terms to one side: \(2x^2 - x - 15 = 0\).
2. Factor the quadratic equation. Look for two numbers that multiply to \(-30\) (the product of \(2\) and \(-15\)) and add to \(-1\) (the coefficient of \(x\)).
3. Once you find the numbers, rewrite the middle term \(-x\) using these numbers to split it into two terms.
4. Factor by grouping. Group the terms into two pairs and factor out the greatest common factor from each pair.
5. Apply the zero-factor property: Set each factor equal to zero and solve for \(x\).

Key Concepts

Zero-Factor Property

The Zero-Factor Property states that if the product of two or more factors equals zero, then at least one of the factors must be zero. This principle is essential for solving polynomial equations, as it allows us to set each factor equal to zero to find the solutions. For example, if we have an equation like (x - 3)(x + 2) = 0, we can conclude that x = 3 or x = -2.

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 using the quadratic formula. In the given problem, we first need to rearrange the equation into standard form before applying the zero-factor property.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This process is crucial for solving equations, as it simplifies the equation to a form where the zero-factor property can be applied. For instance, in the equation 2x^2 - x - 15 = 0, we would factor it into (2x + 5)(x - 3) = 0 to find the values of x that satisfy the equation.