Problem 41 - Elementary Algebra

Question

For the following problems, use the zero-factor property to solve the equations. $$ x(x+8)=0 $$

Step-by-Step Solution

Answer

Question: Use the zero-factor property to solve the quadratic equation x(x+8) = 0. Answer: The roots of the quadratic equation are x=0 and x=-8.

Step 1: Apply the Zero-factor Property

Since x(x+8) = 0, we can apply the zero-factor property. Either x=0 or (x+8)=0.

Step 2: Solve First Equation

The first equation is x=0, which is already solved.

Step 3: Solve Second Equation

The second equation is (x+8)=0. To solve for x, subtract 8 from both sides of the equation: x = -8.

Step 4: Identify the Roots

The roots of the quadratic equation are x=0 and x=-8, which are the values of x that make the equation true.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest degree is two. This means that the variable, usually represented as \(x\), is raised to the power of two. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Quadratic equations can have various forms, and the one in our example, \(x(x + 8) = 0\), is expressed in a factored form. This form is particularly useful because it is easier to solve through the zero-factor property. Solving quadratic equations involves finding the values of \(x\) that satisfy the equation. These solutions are also referred to as the roots of the equation. Understanding quadratic equations is essential as they frequently appear in various mathematical, scientific, and engineering problems. They are foundational in algebra and are crucial for progressing into higher-level mathematics.

Roots

The roots of an equation are the values that satisfy the equation, making it true. For quadratic equations such as \(x(x+8)=0\), the roots are the values of \(x\) that make the equation equal to zero. One of the most straightforward methods for finding the roots of a quadratic equation in factored form is to use the zero-factor property.

• According to this property, if the product of two expressions is zero, then at least one of the expressions must be zero.
• This means, in our case, that either \(x = 0\) or \(x + 8 = 0\).

This rule helps quickly find the values of \(x\) that "break" the equation down to zero. Here, the roots are \(x = 0\) and \(x = -8\). Identifying these roots is critical, as they provide key insights into the behavior of the equation and the graph associated with it.

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and expressions. It is a fundamental area of mathematics, underpinning many other advanced concepts and methods. In algebra, solving equations such as quadratic equations is a way to find unknown variables. With quadratic equations, like the one shown here, the zero-factor property is a critical tool. It simplifies the process of isolating the variable \(x\) to discover its possible values, or roots.Through algebra, we learn methods and strategies, like rearranging terms and factoring, that make solving complex equations manageable. A strong grasp of algebraic principles ensures students can tackle a wide range of mathematical problems using logical and systematic approaches. Moreover, these skills are not only confined to mathematical pursuits but are applicable in fields like physics, engineering, statistics, and computer science, enhancing problem-solving and analytical capabilities.