Selecting the Accurate Number of Solutions

Oct 13, 2025 by windailys.com 119 views

Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand the different types of solutions that can arise from these equations. In this article, we will focus on solving a specific equation and analyzing the types of solutions that can be obtained. We will also examine a given table and select the accurate number of each type of solution.

The Equation

The given equation is:

${ \frac{3}{x} - \frac{x}{x+6} = \frac{18}{x^2 + 6x} }$

This equation involves fractions and rational expressions, and we need to find the values of x that satisfy this equation.

Step 1: Simplify the Equation

To simplify the equation, we can start by finding a common denominator for the fractions. The common denominator is (x)(x+6), so we can rewrite the equation as:

${ \frac{3(x+6) - x(x)}{x(x+6)} = \frac{18}{x(x+6)} }$

Simplifying the numerator, we get:

${ \frac{3x + 18 - x^2}{x(x+6)} = \frac{18}{x(x+6)} }$

Step 2: Eliminate the Fractions

To eliminate the fractions, we can multiply both sides of the equation by the common denominator, x(x+6). This gives us:

${ 3x + 18 - x^2 = 18 }$

Simplifying the equation, we get:

${ -x^2 + 3x + 18 = 0 }$

Step 3: Solve the Quadratic Equation

The resulting equation is a quadratic equation in the form ax^2 + bx + c = 0, where a = -1, b = 3, and c = 18. We can solve this equation using the quadratic formula:

${ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }$

Substituting the values of a, b, and c, we get:

${ x = \frac{-3 \pm \sqrt{3^2 - 4(-1)(18)}}{2(-1)} }$

Simplifying the expression under the square root, we get:

${ x = \frac{-3 \pm \sqrt{9 + 72}}{-2} }$

${ x = \frac{-3 \pm \sqrt{81}}{-2} }$

${ x = \frac{-3 \pm 9}{-2} }$

This gives us two possible solutions:

${ x = \frac{-3 + 9}{-2} = \frac{6}{-2} = -3 }$

${ x = \frac{-3 - 9}{-2} = \frac{-12}{-2} = 6 }$

Analyzing the Solutions

We have obtained two solutions, x = -3 and x = 6. To analyze these solutions, we need to examine the original equation and determine which values of x satisfy the equation.

The Table

The given table is:

Solution Type	Number of Solutions
Real and Rational
Real and Irrational
Complex

We need to select the accurate number of each type of solution based on the solutions we obtained.

Selecting the Accurate Number of Solutions

Based on the solutions we obtained, x = -3 and x = 6, we can analyze the table as follows:

Real and Rational: Since x = 6 is a rational number, we can select 1 as the number of real and rational solutions.
Real and Irrational: Since x = -3 is an irrational number, we can select 0 as the number of real and irrational solutions.
Complex: Since neither x = -3 nor x = 6 is a complex number, we can select 0 as the number of complex solutions.

Conclusion

In this article, we solved a specific equation and analyzed the types of solutions that can be obtained. We also examined a given table and selected the accurate number of each type of solution. The solutions we obtained were x = -3 and x = 6, and we analyzed the table to determine the number of real and rational, real and irrational, and complex solutions.

Final Answer

Based on the analysis, the final answer is:

Solution Type	Number of Solutions
Real and Rational	1
Real and Irrational	0
Complex	0

Introduction

In our previous article, we solved a specific equation and analyzed the types of solutions that can be obtained. We also examined a given table and selected the accurate number of each type of solution. In this article, we will provide a Q&A section to address any questions or concerns that readers may have.

Q: What is the difference between a real and rational solution and a real and irrational solution?

A: A real and rational solution is a solution that is a rational number, which can be expressed as a fraction of two integers. For example, x = 6 is a real and rational solution. On the other hand, a real and irrational solution is a solution that is an irrational number, which cannot be expressed as a fraction of two integers. For example, x = -3 is a real and irrational solution.

Q: How do I determine if a solution is real and rational or real and irrational?

A: To determine if a solution is real and rational or real and irrational, you need to examine the solution and determine if it can be expressed as a fraction of two integers. If it can, then it is a real and rational solution. If it cannot, then it is a real and irrational solution.

Q: What is a complex solution?

A: A complex solution is a solution that involves imaginary numbers, which are numbers that cannot be expressed as a fraction of two integers. Complex solutions typically involve the square root of a negative number.

Q: How do I determine if a solution is complex?

A: To determine if a solution is complex, you need to examine the solution and determine if it involves imaginary numbers. If it does, then it is a complex solution.

Q: What is the significance of the table in the original article?

A: The table in the original article is used to categorize the solutions into real and rational, real and irrational, and complex solutions. This helps to provide a clear understanding of the types of solutions that can be obtained from the equation.

Q: How do I use the table to determine the number of solutions?

A: To use the table to determine the number of solutions, you need to examine the solutions obtained from the equation and categorize them into real and rational, real and irrational, and complex solutions. Then, you can use the table to determine the number of solutions for each category.

Q: What are some common mistakes to avoid when solving equations?

A: Some common mistakes to avoid when solving equations include:

Not simplifying the equation before solving it
Not checking for extraneous solutions
Not using the correct method for solving the equation
Not verifying the solutions

Q: How do I verify the solutions?

A: To verify the solutions, you need to substitute the solutions back into the original equation and check if they satisfy the equation. If they do, then they are valid solutions. If they do not, then they are extraneous solutions.

Conclusion

In this article, we provided a Q&A section to address any questions or concerns that readers may have. We also discussed some common mistakes to avoid when solving equations and provided tips on how to verify the solutions. By following these tips and using the table to categorize the solutions, you can ensure that you are obtaining accurate and reliable results from your equations.

Final Answer

Based on the Q&A section, the final answer is:

A real and rational solution is a solution that is a rational number, which can be expressed as a fraction of two integers.
A real and irrational solution is a solution that is an irrational number, which cannot be expressed as a fraction of two integers.
A complex solution is a solution that involves imaginary numbers, which are numbers that cannot be expressed as a fraction of two integers.
The table in the original article is used to categorize the solutions into real and rational, real and irrational, and complex solutions.
To use the table to determine the number of solutions, you need to examine the solutions obtained from the equation and categorize them into real and rational, real and irrational, and complex solutions.
Some common mistakes to avoid when solving equations include not simplifying the equation before solving it, not checking for extraneous solutions, not using the correct method for solving the equation, and not verifying the solutions.
To verify the solutions, you need to substitute the solutions back into the original equation and check if they satisfy the equation.