Problem 10 - Algebra A Combined Function

Question

Solve each equation. $$ 2 x(x+12)=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -12\).

Step 1: Understand the Structure of the Equation

The given equation is \(2x(x + 12) = 0\). It's a product of two factors equated to zero.

Step 2: Apply the Zero Product Property

If a product of two numbers is zero, at least one of the numbers must be zero. Therefore, set each factor equal to zero: \(2x = 0\) and \(x + 12 = 0\).

Step 3: Solve the First Equation

Solve \(2x = 0\). Divide both sides by 2 to isolate \(x\): \(x = 0\).

Step 4: Solve the Second Equation

Solve \(x + 12 = 0\). Subtract 12 from both sides to isolate \(x\): \(x = -12\).

Step 5: State the Solutions

The solutions to the equation are \(x = 0\) and \(x = -12\). These are the values of \(x\) that satisfy \(2x(x + 12) = 0\).

Key Concepts

Zero Product Property

The zero product property is a fundamental principle when solving quadratic equations. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This concept can seem abstract at first, but it's quite practical when dealing with equations.
Imagine you have a product like \(a \times b = 0\). The zero product property tells us that either \(a = 0\) or \(b = 0\), or both. This is powerful because it allows us to break down seemingly complex equations into simpler parts.
For the equation \(2x(x + 12) = 0\), we apply this property by setting each factor equal to zero:

• Set \(2x = 0\) and solve for \(x\)
• Set \(x + 12 = 0\) and solve for \(x\)

Using this property gives us clear steps to find potential solutions. Recognizing these factors is the first step in solving many quadratic equations.

Solving Equations

Solving equations may initially seem daunting, but breaking them down into steps will help. Each step furthers your understanding and leads you to a solution. For quadratic equations, like \(2x(x + 12) = 0\), this involves both understanding and using properties efficiently.
After applying the zero product property, you'll have smaller, more manageable equations. Here's what you do next:

• For \(2x = 0\), divide both sides by 2 to solve for \(x\). This gives \(x = 0\).
• For \(x + 12 = 0\), subtract 12 from both sides to solve for \(x\). This gives \(x = -12\).

These calculated values, \(x = 0\) and \(x = -12\), are the solutions to the original equation. They satisfy the condition \(2x(x + 12) = 0\) being true, which is what solving an equation ultimately means. Practice applying these steps to different equations to build confidence.

Factoring

Factoring is an essential skill in algebra which simplifies the process of solving equations. It involves rewriting an expression, often a polynomial, as a product of its factors. This makes use of the zero product property easier.
In our equation \(2x(x + 12) = 0\), the expression is already factored. We have two factors: \(2x\) and \((x + 12)\). The product of these two factors results in the original expression.
Factoring is not only about pulling out numbers but also about simplifying how the equation looks and behaves. Given an equation, if it’s not immediately factored, work to express it as a product of linear binomials (or other factors). This step typically involves:

• Identifying common factors
• Rewriting or decomposing terms
• Checking your work by multiplying the factors to ensure they give back the original expression

Mastery of factoring can significantly streamline solving quadratic and higher-order polynomial equations. The practice will hone your ability to spot and manipulate factors efficiently.