Problem 25 - Algebra 1: Concepts and Skills

Question

Use the zero-product property to solve the equation. \(4(c+9)^{2}=0\)

Step-by-Step Solution

Answer

The solution to the equation \(4(c + 9)^{2} = 0\) is \(c = -9\).

Step 1: Simplification

Start by simplifying the equation. We have the equation: \(4(c + 9)^{2} = 0\)This can be interpreted as 4 multiplied by \((c + 9)^{2}\). The problem asks to use the zero product property, meaning that one of the factors must be zero. The factors here are 4 and \((c + 9)^{2}\).

Step 2: Apply the Zero-Product Property

Since the product of 4 and \((c + 9)^{2}\) equals zero, and 4 obviously is not zero, therefore we know that \((c + 9)^{2} = 0\). Now, identify the terms within the parentheses that would result in a product of zero when squared.

Step 3: Solve the Equation

To solve the equation \((c + 9)^{2} = 0\), take the square root of both sides. Hence, \(c + 9 = 0\). Solving for \(c\), we subtract 9 from both sides, which gives us \(c = -9\).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two. They usually appear in the form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. These equations are fundamental in algebra and arise in various real-world applications. If you encounter a term squared, like \((c+9)^2\), you are dealing with a quadratic equation.

The degree of a polynomial is important because it tells us how many solutions, or roots, the equation can have. For quadratics, there can be up to two solutions. These solutions can be found using several methods, such as factoring, completing the square, or the quadratic formula. However, sometimes the equation is set up so that methods like the zero-product property make solving quicker and easier.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In algebra, equations are like balance scales. When solving, the goal is to isolate the variable (usually represented by a letter, like \(c\) or \(x\)) on one side of the equation. This involves manipulating the equation through various operations.

For the equation \(4(c + 9)^2 = 0\), the zero-product property can be used. This property states that if a product of factors equals zero, then at least one of the factors must be zero. Therefore, we look at each factor separately to find the solution. Here, \(4\) (a constant term) does not influence the equation being zero because it is not zero itself. Thus, we focus on \((c + 9)^2 = 0\) to find our solution by taking the square root of both sides, which simplifies the solving process.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations like addition, subtraction, multiplication, and division. They form the building blocks for equations. An expression like \((c + 9)^2\) is a perfect example. The expression inside the parentheses, \((c + 9)\), is considered a binomial because it contains two terms.

When expressions are squared, it often simplifies the process of solving equations, especially when looking for factors that make the expression equal zero, as in the zero-product property. Understanding how to manipulate algebraic expressions efficiently is crucial for solving more complex equations. Recognizing these patterns becomes second nature as one becomes more familiar with algebra, allowing problems to be approached with confidence.