Substituting the Simplified Numerator

Oct 13, 2025 by windailys.com 145 views

Introduction

Solving equations involving fractions can be a challenging task in mathematics. In this article, we will focus on solving the given equation $\frac{x-3}{x} - \frac{3}{x+1} + \frac{3}{x^2+x} = 0$ . To solve this equation, we need to simplify it and then use algebraic techniques to find the value of $x$ . The main keywords in this article are solving equations, fractions, and algebraic techniques.

Simplifying the Equation

The given equation is $\frac{x-3}{x} - \frac{3}{x+1} + \frac{3}{x^2+x} = 0$ . To simplify this equation, we can start by finding a common denominator for the three fractions. The common denominator is $x(x+1)(x^2+x)$ .

import sympy as sp
x = sp.symbols('x')
equation = (x-3)/x - 3/(x+1) + 3/(x**2+x)
simplified_equation = sp.simplify(equation)

Combining the Fractions

After finding the common denominator, we can combine the fractions by adding and subtracting their numerators.

$\frac{x-3}{x} - \frac{3}{x+1} + \frac{3}{x^2+x} = \frac{(x-3)(x+1)(x^2+x) - 3x(x^2+x) + 3x}{x(x+1)(x^2+x)}$

Expanding the Numerator

To simplify the numerator, we can expand it by multiplying the factors.

$(x-3)(x+1)(x^2+x) - 3x(x^2+x) + 3x = x^4 + x^3 - 3x^3 - 3x^2 + 3x^3 + 3x^2 - 3x^2 - 3x + 3x$

Simplifying the Numerator

After expanding the numerator, we can simplify it by combining like terms.

$x^4 + x^3 - 3x^3 - 3x^2 + 3x^3 + 3x^2 - 3x^2 - 3x + 3x = x^4 - 3x^2$

Substituting the Simplified Numerator

Now that we have simplified the numerator, we can substitute it back into the equation.

$\frac{x^4 - 3x^2}{x(x+1)(x^2+x)} = 0$

Factoring the Numerator

To make it easier to solve the equation, we can factor the numerator.

$x^4 - 3x^2 = x^2(x^2 - 3)$

Substituting the Factored Numerator

Now that we have factored the numerator, we can substitute it back into the equation.

$\frac{x^2(x^2 - 3)}{x(x+1)(x^2+x)} = 0$

Canceling Common Factors

Since the numerator and denominator have a common factor of $x^2$ , we can cancel it out.

$\frac{x^2 - 3}{x(x+1)(x^2+x)} = 0$

Finding the Value of $x$

To find the value of $x$ , we can set the numerator equal to zero and solve for $x$ .

$x^2 - 3 = 0$

Solving for $x$

Now that we have set the numerator equal to zero, we can solve for $x$ .

$x^2 = 3$

$x = \pm \sqrt{3}$

Conclusion

In this article, we have solved the equation $\frac{x-3}{x} - \frac{3}{x+1} + \frac{3}{x^2+x} = 0$ using algebraic techniques. We simplified the equation by finding a common denominator and combining the fractions. We then expanded and simplified the numerator, factored it, and canceled out common factors. Finally, we found the value of $x$ by setting the numerator equal to zero and solving for $x$ . The main keywords in this article are solving equations, fractions, and algebraic techniques.

Discussion

The equation $\frac{x-3}{x} - \frac{3}{x+1} + \frac{3}{x^2+x} = 0$ is a type of rational equation, which involves fractions with variables in the numerator and denominator. Solving rational equations can be challenging, but by using algebraic techniques, we can simplify the equation and find the value of $x$ . In this article, we have used the following algebraic techniques:

Finding a common denominator
Combining fractions
Expanding and simplifying the numerator
Factoring the numerator
Canceling common factors
Setting the numerator equal to zero and solving for $x$

These algebraic techniques are essential in solving rational equations and can be applied to a wide range of mathematical problems.

Final Answer

The final answer is $\boxed{\pm \sqrt{3}}$ .

Introduction

In our previous article, we solved the equation $\frac{x-3}{x} - \frac{3}{x+1} + \frac{3}{x^2+x} = 0$ using algebraic techniques. In this article, we will answer some frequently asked questions (FAQs) related to solving this equation.

Q1: What is the main concept behind solving this equation?

A1: The main concept behind solving this equation is to simplify it by finding a common denominator and combining the fractions. We then expand and simplify the numerator, factor it, and cancel out common factors. Finally, we set the numerator equal to zero and solve for $x$ .

Q2: Why is it important to find a common denominator?

A2: Finding a common denominator is important because it allows us to combine the fractions and simplify the equation. Without a common denominator, we would not be able to simplify the equation and would have to deal with multiple fractions.

Q3: How do we expand and simplify the numerator?

A3: To expand and simplify the numerator, we multiply the factors and combine like terms. This involves multiplying the terms in the numerator and then combining the like terms.

Q4: Why do we factor the numerator?

A4: We factor the numerator to make it easier to solve the equation. Factoring the numerator allows us to set the numerator equal to zero and solve for $x$ .

Q5: What is the final answer to the equation?

A5: The final answer to the equation is $\boxed{\pm \sqrt{3}}$ .

Q6: Can we use other algebraic techniques to solve this equation?

A6: Yes, we can use other algebraic techniques to solve this equation. However, the techniques we used in this article are some of the most common and effective methods for solving rational equations.

Q7: How do we know when to use algebraic techniques to solve an equation?

A7: We know when to use algebraic techniques to solve an equation when the equation involves fractions or other types of rational expressions. Algebraic techniques are particularly useful when dealing with rational equations.

Q8: Can we apply the techniques used in this article to other types of equations?

A8: Yes, we can apply the techniques used in this article to other types of equations. The techniques we used in this article are general and can be applied to a wide range of mathematical problems.

Q9: What are some common mistakes to avoid when solving rational equations?

A9: Some common mistakes to avoid when solving rational equations include:

Not finding a common denominator
Not expanding and simplifying the numerator
Not factoring the numerator
Not canceling common factors
Not setting the numerator equal to zero and solving for $x$

Q10: How can we practice solving rational equations?

A10: We can practice solving rational equations by working through examples and exercises. We can also use online resources and math software to help us practice and improve our skills.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to solving the equation $\frac{x-3}{x} - \frac{3}{x+1} + \frac{3}{x^2+x} = 0$ . We have discussed the main concept behind solving this equation, the importance of finding a common denominator, and how to expand and simplify the numerator. We have also answered questions about factoring the numerator, canceling common factors, and setting the numerator equal to zero and solving for $x$ . Finally, we have discussed some common mistakes to avoid when solving rational equations and how to practice solving rational equations.

Final Answer

The final answer is $\boxed{\pm \sqrt{3}}$ .