Problem 25 - Algebra A Combined Function

Question

Solve. $$ x^{2}+20 x=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -20\).

Step 1: Identify the structure of the equation

The given equation is of the form \(x^2 + ax = 0\), which is a quadratic equation with two terms, \(x^2\) and \(20x\). To solve it, you can factor out the common variable, \(x\).

Step 2: Factor the equation

Look for common factors in the equation. Both terms \(x^2\) and \(20x\) have \(x\) as a common factor. Factoring out \(x\), the equation becomes \(x(x + 20) = 0\).

Step 3: Use the Zero Product Property

According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, set each factor equal to zero: \(x = 0\) and \(x + 20 = 0\).

Step 4: Solve for x

Solve each equation obtained from the Zero Product Property. 1. \(x = 0\) is already solved. 2. For \(x + 20 = 0\), subtract 20 from both sides to get \(x = -20\).

Step 5: List the solutions

The solutions to the equation \(x^2 + 20x = 0\) are \(x = 0\) and \(x = -20\). These are the values of \(x\) that satisfy the original equation.

Key Concepts

Factoring

Factoring is a crucial method for solving quadratic equations, and it involves breaking down an equation into simpler components.
In the context of the given equation, our goal is to identify common factors in the expression.
The equation, \(x^2 + 20x = 0\), consists of two terms: \(x^2\) and \(20x\). Here, you can readily see that both terms have a common factor of \(x\).

• Extracting \(x\) from each term simplifies the equation.
• By factoring, you transform the equation into \(x(x + 20) = 0\).

For beginners, imagining factoring as "reverse distribution" might help; it's like pulling a common piece out from each term.
Knowing how to factor is foundational for solving more complex algebraic equations.

Zero Product Property

The Zero Product Property is a fundamental principle in algebra that aids in solving factored equations.
This property states that if a product of two numbers is zero, at least one of the numbers must be zero.
In simpler terms:

• If \(a \times b = 0\), then either \(a = 0\) or \(b = 0\) (or both).

Applying this to our factored equation \(x(x + 20) = 0\):

• We deduce that either \(x = 0\) or \(x + 20 = 0\).

These separate equations can be solved individually.
The first equation \(x = 0\) is straightforward. The second equation, \(x + 20 = 0\), involves a simple subtraction to isolate \(x\).
Zero Product Property is a powerful tool that simplifies the process of finding solutions.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols.
It provides a framework for solving equations and involves working with variables to express mathematical relationships.
In our example with the quadratic equation \(x^2 + 20x = 0\), algebra involves:

• Recognizing the type of equation
• Using techniques such as factoring and applying the Zero Product Property
• Solving the resulting equations to find the values of \(x\)

Algebra is essential because it enables problem-solving beyond simple arithmetic, allowing for the manipulation of expressions and equations that represent real-world scenarios.
Mastery of algebra is vital for understanding more advanced mathematical concepts and applications.