Solve each equation in Exercises 1 - 14 by factoring. $3x^2+12x=0$

Verified step by step guidance

1. Start by writing the equation: \$3x^2 + 12x = 0$.
2. Factor out the greatest common factor (GCF) from both terms. The GCF of \$3x^2\( and \)12x\( is \)3x\(, so factor it out: \)3x(x + 4) = 0$.
3. Apply the Zero Product Property, which states that if a product of two factors is zero, then at least one of the factors must be zero. Set each factor equal to zero: \$3x = 0\( and \)x + 4 = 0$.
4. Solve each equation separately: For \$3x = 0\(, divide both sides by 3 to get \)x = 0\(. For \)x + 4 = 0\(, subtract 4 from both sides to get \)x = -4$.
5. Write the solution set combining both values: \(x = 0\) and \(x = -4\).

Key Concepts

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of simpler expressions, typically binomials or monomials. For example, 3x² + 12x can be factored by taking out the greatest common factor (GCF), which simplifies solving the equation.

Greatest Common Factor (GCF)

The GCF is the largest expression that divides all terms of a polynomial without a remainder. Identifying and factoring out the GCF simplifies the equation, making it easier to solve by setting each factor equal to zero.

Zero Product Property

This property states that if the product of two factors equals zero, then at least one of the factors must be zero. After factoring the equation, set each factor equal to zero to find the solutions of the equation.