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

Verified step by step guidance

1. Start by expanding the right-hand side of the equation: \((3x + 2)(x - 1)\).
2. Distribute each term in the first binomial across the second binomial: \(3x(x - 1) + 2(x - 1)\).
3. Simplify the expression by performing the multiplication: \(3x^2 - 3x + 2x - 2\).
4. Combine like terms to get: \(3x^2 - x - 2\).
5. Set the equation to zero by moving all terms to one side: \(3x^2 - x - 2 - 7 + 7x = 0\), then simplify to \(3x^2 + 6x - 9 = 0\).

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 + 6 can be factored into (x - 2)(x - 3).

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 crucial when solving equations that have been factored, as it allows us to find the values of the variable that satisfy the equation. For instance, if (x - 2)(x - 3) = 0, then x must be either 2 or 3.

Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. They can be represented in the form ax + b = 0, where a and b are constants. Understanding linear equations is vital for solving more complex equations, as they often serve as the foundation for algebraic manipulation and problem-solving techniques.