Solve each equation in Exercises 1 - 14 by factoring. 2x(x - 3) = 5x^2 - 7x

Verified step by step guidance

1. Start by expanding the left side of the equation: \(2x(x - 3) = 2x^2 - 6x\).
2. Rearrange the equation to bring all terms to one side: \(2x^2 - 6x = 5x^2 - 7x\) becomes \(0 = 5x^2 - 7x - 2x^2 + 6x\).
3. Combine like terms: \(0 = 3x^2 - x\).
4. Factor out the greatest common factor from the right side: \(0 = x(3x - 1)\).
5. Set each factor equal to zero and solve for \(x\): \(x = 0\) or \(3x - 1 = 0\).

Key Concepts

Factoring

Factoring is the process of breaking down an expression into a product of simpler expressions, or factors, that when multiplied together yield the original expression. This technique is essential in solving polynomial equations, as it allows us to set each factor equal to zero to find the solutions. For example, the expression x^2 - 5x can be factored into x(x - 5).

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically in the form ax^2 + bx + c = 0. In the context of factoring, we often rearrange the equation to set it to zero and then factor it to find the values of x that satisfy the equation. Understanding the standard form and properties of quadratic equations is crucial for effective problem-solving.

Zero Product Property

The Zero Product Property states that if the product of two or more factors equals zero, at least one of the factors must be zero. This principle is fundamental when solving factored equations, as it allows us to find the roots of the equation by setting each factor to zero. For instance, if we have (x - 2)(x + 3) = 0, we can conclude that x = 2 or x = -3.