Problem 98 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Both 0 and \(-\pi\) are solutions of the equation \(x(x+\pi)=0\)

Step-by-Step Solution

Answer

Yes, both 0 and \(-\pi\) are solutions of the given equation \(x(x+\pi)=0\).

Step 1: Test with 0

Firstly, substitute 0 into the given equation \(x(x+\pi)=0\) to get \(0(0+\pi) = 0 \cdot \pi = 0\). This indeed equals 0, which means 0 is a solution of the equation.

Step 2: Test with -π

Secondly, substitute -π into the equation \(x(x+\pi)=0\). You'll get, \(-\pi(-\pi+\pi) = -\pi \cdot 0 = 0\). This again equals 0, which means -π is a solution of the equation.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that relate one or more unknown variables using algebraic expressions. At their core, they set one expression equal to another. In solving these equations, the goal is to find the values of the unknown variables that make the equation true.

For example, in the equation \(x(x+\pi)=0\), we have one variable \(x\) that we're trying to solve for. This is a quadratic equation because it involves \(x\) squared. The equation is set to zero, which means we're looking for the values of \(x\) that make this product zero.

Remember:

• If a product of multiple terms is zero, at least one of those terms must be zero. This property helps in solving equations like the one in this exercise.

Problem Solving

Problem solving in mathematics involves a systematic approach to find solutions to given problems. It requires understanding the problem, devising a plan, carrying out the plan, and evaluating the solution.

In our exercise, the problem is to determine if 0 and \(-\pi\) are solutions to \(x(x+\pi)=0\). We use a trial and error method where we substitute each value into the equation to see if it satisfies the equation.

Steps to remember:

• Understand the problem by identifying what is being asked.
• Substitute the given values one at a time into the equation.
• Calculate the result of each substitution to verify if it equals zero.
• If both satisfy the equation, both are solutions.

Substitution Method

The substitution method is a key technique in solving algebraic equations. It involves replacing variables with specific values to see if those values satisfy the equation.

For the equation \(x(x+\pi)=0\):

• We first substitute \(x=0\). This gives us \(0(0+\pi)=0\), which simplifies to 0, confirming that 0 is a solution.
• Next, we substitute \(x=-\pi\). This gives \(-\pi(-\pi+\pi)=0\), which simplifies again to 0, confirming that \(-\pi\) is a solution.

Substitution is powerful because:

• It directly verifies if a given value solves the equation.
• Helps to easily check multiple solutions for correctness.