Problem 349 - Elementary Algebra

Question

Solve. \((x+8)(x-3)=0\)

Step-by-Step Solution

Answer

x = -8 and x = 3

Step 1 - Apply the Zero Product Property

To solve \( (x+8)(x-3)=0 \), use the Zero Product Property which states if a product of two factors is zero, then at least one of the factors must be zero. Therefore, set each factor equal to zero: \( x+8=0 \) and \( x-3=0 \).

Step 2 - Solve for x

Solve each equation from Step 1 separately. \( x+8=0 \) simplifies to \[ x = -8 \]. \( x-3=0 \) simplifies to \[ x = 3 \].

Step 3 - Combine the Solutions

Combine the solutions from Step 2. Therefore, the solutions to the equation \( (x+8)(x-3)=0 \) are \ x = -8 \ and \ x = 3 \.

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations of the form a x^2 + b x + c = 0 . They have a variable raised to the power of 2. When solving these equations, our goal is to find the values of the variable (often x) that make the equation true. In this given exercise, the equation was already factored into (x+8)(x-3)=0, which makes it straightforward to apply specific techniques for solving. Just remember the overall goal, which is to solve for the variable that satisfies the equation.

Factoring Technique

The Factoring Technique is one of the methods used to solve quadratic equations, especially when the equation can easily be written as a product of simpler expressions. In the initial problem, we saw (x+8)(x-3)=0. This equation is already factored, meaning it is already written as a product of two binomials. By factoring the quadratic equation into such expressions, we can utilize the Zero Product Property to solve it. Factoring simplifies the process as it breaks the quadratic equation down into smaller parts that are easier to solve. For many quadratic equations, the first step is to factor them into a product of binomials.

Algebraic Solutions

Algebraic solutions involve applying algebraic principles to find the values of the variable that satisfy the equation. In our problem, after factoring, we used the Zero Product Property, which states that if a product of two numbers is zero, then at least one of the numbers must be zero. Here’s how it works:

• Given (x+8)(x-3)=0, we set each factor to zero: x+8=0 and x-3=0.
• Next, solve each of these simpler equations separately:
• For x+8=0, we subtract 8 from both sides to get x=-8.
• For x-3=0, we add 3 to both sides to get x=3.

Combining these results, we get the solutions x=-8 and x=3. This method of breaking down a complex problem into manageable steps is fundamental in algebra.