Problem 28 - Algebra 1: Concepts and Skills

Question

Use the zero-product property to solve the equation. \((d+6)(3 d-4)=0\)

Step-by-Step Solution

Answer

The solutions to the equation \((d+6)(3d-4)=0\) are \(d = -6\) and \(d = \frac{4}{3}\)

Step 1: Apply the Zero-Product Property

Since the equation is already in a product form and equals to zero, apply the zero-product property, which states that if the product of two terms equals zero, then at least one of the terms must be zero. So we set each term to zero: \(d+6 = 0\) and \(3d-4 = 0\)

Step 2: Solve the equations

To solve the first equation \(d+6 = 0\) for \(d\), subtract 6 from both sides to get \(d = -6\). To solve the second equation \(3d-4 = 0\) for \(d\), add 4 to both sides to get \(3d = 4\), then divide by 3 to get \(d = \frac{4}{3}\)

Key Concepts

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. In our exercise, we are solving the equation \((d+6)(3d-4)=0\). Here, the equation is already in a nice form for solving because it is the product of two expressions. When using the zero-product property, we recognize that if the product of two things is zero, one of those things has to be zero. So, we set each part of our product equal to zero:

• \(d+6 = 0\)
• \(3d-4 = 0\)

Once we set each part equal to zero, we solve these simpler equations one by one. This gives us the solutions for \(d\). Each separate equation is like a small riddle that we solve to find parts of our mystery answer. This method is straightforward and only requires simple algebraic manipulations like addition and division.

Factoring

Factoring is the process of breaking down an expression into a product of its factors. It is like figuring out which smaller parts multiply together to give the whole expression. In the context of equations, like \((d+6)(3d-4)=0\), the expression is already factored. Factoring is crucial because it allows us to apply the zero-product property effectively. Consider the expression \((d+6)\) and \((3d-4)\); these are the factors.

• The factor \((d+6)\) means \(d\) has a shift of 6 to balance it back to zero.
• The factor \((3d-4)\) requires manipulating both the multiplication and subtraction to ease to zero.

By breaking an expression into factors, we convert a possibly complex polynomial into a set of simpler linear equations. Factoring is often the first step in solving polynomial equations, making it a cornerstone concept in algebra.

Algebraic Expressions

Algebraic expressions are made up of variables, numbers, and operations. In the equation \((d+6)(3d-4)=0\), each part \((d+6)\) and \((3d-4)\) is an algebraic expression. Understanding what an algebraic expression is helps us analyze and manipulate it to solve equations. These expressions can look like:

• Terms: individual parts separated by plus or minus signs, such as \(d\), \(3d\), or numbers like \(6\) and \(-4\).
• Factors: parts that are multiplied together, like \(d+6\) is a factor of the larger product.
• Coefficients: numbers that multiply variables, here \(3\) is the coefficient of \(d\) in \(3d-4\).

Algebraic expressions must be identified correctly to solve any algebra problem. Learning to manipulate these expressions—through operations like addition, subtraction, and distribution (factoring)—is key to mastering algebra.