Solve each equation in Exercises 58–59 by factoring. 2x^2 +15x = 8

Verified step by step guidance

1. First, move all terms to one side of the equation to set it to zero: $2x^2+15x-8=0$.
2. Next, identify the coefficients for the quadratic equation $a{x}^2+bx+c=0$, where $a=2$, $b=15$, and $c=-8$.
3. Look for two numbers that multiply to $a\times c=2\times (-8)=-16$ and add up to $b=15$.
4. Once you find these numbers, use them to split the middle term $15x$ into two terms, and rewrite the equation as $2x^2+mx+nx-8=0$, where $m$ and $n$ are the numbers found.
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 quadratic in the form ax^2 + bx + c = 0, we look for two numbers that multiply to ac and add to b. This technique simplifies solving the equation by setting each factor to zero.

Zero Product Property

The Zero Product Property states that if the product of two factors equals zero, at least one of the factors must be zero. This principle is crucial when solving factored equations, as it allows us to set each factor equal to zero and solve for the variable, leading to the possible solutions of the original equation.

Rearranging Equations

Rearranging equations involves moving terms from one side of the equation to the other to isolate the variable. In the context of the given equation, we first need to set it to zero by moving all terms to one side, transforming it into a standard quadratic form. This step is essential for applying factoring and the Zero Product Property effectively.