Problem 17 - Algebra A Combined Function

Question

Solve each equation. $$ (x+0.2)(x+1.5)=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -0.2\) and \(x = -1.5\).

Step 1: Understand the Zero Product Property

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. This principle can be used to find the solutions for the quadratic equation \((x+0.2)(x+1.5)=0\).

Step 2: Set Each Factor to Zero

Using the zero product property, set each factor in the equation equal to zero: \(x + 0.2 = 0\) and \(x + 1.5 = 0\).

Step 3: Solve First Equation for x

Solve the first equation \(x + 0.2 = 0\) by isolating \(x\). Subtract 0.2 from both sides to get \(x = -0.2\).

Step 4: Solve Second Equation for x

Solve the second equation \(x + 1.5 = 0\) by isolating \(x\). Subtract 1.5 from both sides to get \(x = -1.5\).

Step 5: Write the Solution Set

Combine the solutions from steps 3 and 4 to write the solution set. The equation \((x+0.2)(x+1.5)=0\) has solutions \(x = -0.2\) and \(x = -1.5\).

Key Concepts

Zero Product Property

The zero product property is a fundamental rule in algebra that helps solve equations involving products of expressions. It says that if you multiply two numbers together and the result is zero, then at least one of the numbers must be zero. This means:

• If \( ab = 0 \), then \( a = 0 \) or \( b = 0 \).

This principle is especially useful in solving quadratic equations in factored form. If you encounter an equation like \[(x+0.2)(x+1.5)=0,\]you can directly apply the zero product property to find potential solutions. By setting each factor (\(x + 0.2\) and \(x + 1.5\)) equal to zero, you break down the problem into simpler, solvable equations.

Factoring

Factoring involves rewriting a mathematical expression as a product of its factors, or simpler terms, that, when multiplied together, give back the original expression. It is a crucial step when solving quadratic equations because it allows us to apply the zero product property. Using factoring, you can transform complex expressions into manageable pieces.

• For example, if you have an equation like \( ax^2 + bx + c = 0 \), your goal is to express it as \((mx + n)(px + q) = 0\).
• The factored form exposes the roots, or solutions, of the equation where the expression equals zero.

In the provided problem, the expression is already factored as \[(x+0.2)(x+1.5)=0.\]Thus, applying the zero product property becomes straightforward. Remember, the skill of factoring is not just about solving equations but also about understanding the structure and relationships of polynomials.

Solution Sets

A solution set is a group of values that satisfy a given equation. After you've factored an equation and applied the zero product property, the next step is to find the solutions, or the values of \( x \), that make the equation true. Each solution corresponds to a scenario where each factor equals zero.
For the problem\[(x+0.2)(x+1.5)=0,\]you have the equations \(x + 0.2 = 0\) and \(x + 1.5 = 0\).

• Solving \(x + 0.2 = 0\), you find \(x = -0.2\).
• Solving \(x + 1.5 = 0\), you find \(x = -1.5\).

Thus, the solution set for the equation is \( \{-0.2, -1.5\} \). This encompasses all possible values of \( x \) where the original equation holds true. Understanding solution sets help in visualizing the points where a graph of the equation will intersect the x-axis.