Problem 41 - Algebra for College Students

Question

Explain what it means to say that the equation \((x+3)^{2}=\) 0 has a solution of \(-3\) with a multiplicity of two.

Step-by-Step Solution

Answer

The equation has a root of \(-3\), repeated twice as it's squared.

Step 1: Understanding the Equation

The equation given is \((x+3)^{2}=0\). This is a quadratic equation that is written in its factored form. The expression \((x+3)\) is squared, which means it is multiplied by itself.

Step 2: Identifying the Root

To solve the equation \((x+3)^{2}=0\), we need to find the value of \(x\) that makes \((x+3)^{2}\) equal to zero. Since it is a squared term, the expression inside the square must be zero. Set \(x+3 = 0\) to find the root.

Step 3: Solving for the Root

Solving \(x+3 = 0\), we subtract 3 from both sides to isolate \(x\). This gives us \(x = -3\). Thus, the root of the equation is \(-3\).

Step 4: Understanding Multiplicity

The power of the factor \((x+3)\) in \((x+3)^{2}\) indicates the multiplicity of the root. Here, \((x+3)\) is squared, and therefore the root \(-3\) appears twice. This means \(-3\) is a repeated root, with a multiplicity of two.

Key Concepts

Factored Form

When we talk about the factored form of a quadratic equation, we mean expressing the equation as a product of its factors. Imagine the equation as broken down into simpler parts that, when multiplied, give you the original equation. For example, in the quadratic equation \((x+3)^2 = 0\), it is already presented in its factored form. This means that the expression \((x+3)\) is repeated and multiplied by itself.

Why is factored form useful? It simplifies solving the equation by showing us the individual components that make up the zero. Once you have an equation in this form, it's much easier to identify potential solutions, or roots, because you can set each part equal to zero and solve for \(x\).

In summary:

• The factored form breaks the equation into a product of factors.
• It's easier to find the equation's roots.
• It directly shows the multiplicity of roots.

Roots of Equations

Roots of an equation are the values for which the equation equals zero. These values satisfy the equation. In our example, solving \((x+3)^2 = 0\), we focus on making the expression equal to zero to find the roots.

To find a root, you take the factor \((x+3)\) in the equation and set it to zero: \(x + 3 = 0\). Solving this simple equation, we subtract 3 from both sides and find \(x = -3\). So, the root of the equation is -3, meaning if you replace \(x\) with -3 in the equation, the original expression equals zero.

Here's the process in steps:

• Set the factor to zero: \(x+3=0\).
• Solve for \(x\).
• The result is the root of the equation.

Remember, roots are essential because they are the solutions to the equation. They tell us where the graph of the equation would touch or intersect the x-axis.

Multiplicity of a Root

Multiplicity refers to the number of times a particular root appears in the equation. In other words, it's the number of identical solutions resulting from the equation. In the given example of \((x+3)^2 = 0\), the factor \((x+3)\) is squared, indicating that the root \(-3\) has a multiplicity of two.

So, what does a multiplicity of two mean? It means the root \(-3\) is repeated twice in the equation. When you solve it, you are essentially finding \(-3\) two times as a solution. In the context of graphs, a root with an even multiplicity, like two, means the graph "touches" the x-axis at the root point but doesn’t cross it.

Why is understanding multiplicity important?

• It shows how many times a root is a solution.
• A higher multiplicity affects the shape of the graph at that root.
• In equations, it helps predict behavior near the solution.

In this way, the multiplicity provides deeper insights into the nature of the equation's solutions.