Solve each equation in Exercises 1 - 14 by factoring. 3x^2 - 2x = 8

Verified step by step guidance

1. Start by moving all terms to one side of the equation to set it equal to zero: $3x^2-2x-8=0$.
2. Identify the quadratic equation in the form $a{x}^2+bx+c=0$, where $a=3$, $b=-2$, and $c=-8$.
3. Look for two numbers that multiply to $a\cdot c=3\times (-8)=-24$ and add to $b=-2$.
4. Once the numbers are found, rewrite the middle term $-2x$ using these numbers to split it into two terms.
5. Factor by grouping: group the terms into two pairs and factor out the greatest common factor from each pair, then factor out the common binomial factor.

Key Concepts

Factoring Quadratic Equations

Factoring is the process of breaking down a quadratic equation into simpler expressions that can be multiplied to yield the original equation. In the case of a standard quadratic equation in the form ax^2 + bx + c = 0, factoring involves finding two binomials that multiply to give the quadratic. This method is essential for solving equations, as it allows us to set each factor equal to zero and solve for the variable.

Setting the Equation to Zero

To solve an equation by factoring, it is crucial to rearrange the equation so that one side equals zero. This is typically done by moving all terms to one side of the equation. For example, in the equation 3x^2 - 2x - 8 = 0, we isolate the quadratic expression, which allows us to apply the factoring method effectively and find the values of x that satisfy the equation.

Zero Product Property

The Zero Product Property states that if the product of two factors equals zero, then at least one of the factors must be zero. This principle is fundamental when solving factored equations, as it allows us to set each factor equal to zero and solve for the variable. For instance, if we factor the equation into (ax + b)(cx + d) = 0, we can conclude that either ax + b = 0 or cx + d = 0 must hold true.