Solve each equation using the zero-factor property. 2x² - x = 15

Verified step by step guidance

1. First, rewrite the equation so that one side equals zero. Start with the given equation: \$2x^2 - x = 15\(. Subtract 15 from both sides to get: \)2x^2 - x - 15 = 0$.
2. Next, factor the quadratic expression \$2x^2 - x - 15\(. Look for two binomials of the form \)(ax + b)(cx + d)\( that multiply to \)2x^2 - x - 15$.
3. Use the zero-factor property, which states that if \(AB = 0\), then either \(A = 0\) or \(B = 0\). Set each factor equal to zero separately.
4. Solve each resulting linear equation for \(x\). This will give you the possible solutions to the original equation.
5. Finally, check your solutions by substituting them back into the original equation to verify they satisfy it.

Key Concepts

Zero-Factor Property

The zero-factor property states that if the product of two factors is zero, then at least one of the factors must be zero. This property is essential for solving polynomial equations by factoring, as it allows us to set each factor equal to zero and solve for the variable.

Rearranging Equations to Standard Form

Before applying the zero-factor property, the equation must be set to zero on one side. This involves moving all terms to one side to form a quadratic equation in standard form (ax^2 + bx + c = 0), which is necessary for factoring or other solution methods.

Factoring Quadratic Expressions

Factoring involves expressing a quadratic expression as a product of two binomials. Recognizing common factoring techniques, such as factoring out the greatest common factor or using methods like grouping or the quadratic formula, is crucial to break down the equation for applying the zero-factor property.