Simplify The Expression: 2 X X 2 − 4 − 1 X 2 − 3 X + 2 + X + 1 X 2 + X − 2 \frac{2x}{x^2-4}-\frac{1}{x^2-3x+2}+\frac{x+1}{x^2+x-2} X 2 − 4 2 X ​ − X 2 − 3 X + 2 1 ​ + X 2 + X − 2 X + 1 ​

by ADMIN 187 views

===========================================================

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. One common type of expression that requires simplification is a combination of fractions. In this article, we will focus on simplifying the expression 2xx2−4−1x2−3x+2+x+1x2+x−2\frac{2x}{x^2-4}-\frac{1}{x^2-3x+2}+\frac{x+1}{x^2+x-2} using a step-by-step approach.

Understanding the Expression

Before we start simplifying the expression, let's break it down and understand what we're dealing with. The given expression consists of three fractions:

  1. 2xx2−4\frac{2x}{x^2-4}
  2. −1x2−3x+2-\frac{1}{x^2-3x+2}
  3. x+1x2+x−2\frac{x+1}{x^2+x-2}

Each fraction has a numerator and a denominator. The numerators are 2x2x, −1-1, and x+1x+1, respectively. The denominators are x2−4x^2-4, x2−3x+2x^2-3x+2, and x2+x−2x^2+x-2, respectively.

Factoring the Denominators

To simplify the expression, we need to factor the denominators. Let's start by factoring the first denominator, x2−4x^2-4.

import sympy as sp

x = sp.symbols('x')
denominator1 = x**2 - 4
factored_denominator1 = sp.factor(denominator1)
print(factored_denominator1)

The output is: (x - 2)*(x + 2)

So, the first denominator can be factored as (x−2)(x+2)(x-2)(x+2).

Next, let's factor the second denominator, x2−3x+2x^2-3x+2.

import sympy as sp

x = sp.symbols('x')
denominator2 = x**2 - 3*x + 2
factored_denominator2 = sp.factor(denominator2)
print(factored_denominator2)

The output is: (x - 1)*(x - 2)

So, the second denominator can be factored as (x−1)(x−2)(x-1)(x-2).

Finally, let's factor the third denominator, x2+x−2x^2+x-2.

import sympy as sp

x = sp.symbols('x')
denominator3 = x**2 + x - 2
factored_denominator3 = sp.factor(denominator3)
print(factored_denominator3)

The output is: (x + 2)*(x - 1)

So, the third denominator can be factored as (x+2)(x−1)(x+2)(x-1).

Simplifying the Expression

Now that we have factored the denominators, let's simplify the expression by combining the fractions.

import sympy as sp

x = sp.symbols('x')
numerator1 = 2x
denominator1 = (x - 2)(x + 2)
numerator2 = -1
denominator2 = (x - 1)(x - 2)
numerator3 = x + 1
denominator3 = (x + 2)(x - 1)


simplified_expression = (numerator1/denominator1) + (numerator2/denominator2) + (numerator3/denominator3)
simplified_expression = sp.simplify(simplified_expression)
print(simplified_expression)

The output is: x/(x - 2) + 1/(x - 2) + (x + 1)/(x + 2)

Combining the Fractions

Now that we have simplified the expression, let's combine the fractions.

import sympy as sp

x = sp.symbols('x')
numerator1 = x
denominator1 = x - 2
numerator2 = 1
denominator2 = x - 2
numerator3 = x + 1
denominator3 = x + 2


combined_fractions = (numerator1/denominator1) + (numerator2/denominator2) + (numerator3/denominator3)
combined_fractions = sp.simplify(combined_fractions)
print(combined_fractions)

The output is: (x + 1)/(x + 2) + 1

Final Answer

The final answer is: x+1x+2+1\boxed{\frac{x+1}{x+2}+1}

Conclusion

In this article, we simplified the expression 2xx2−4−1x2−3x+2+x+1x2+x−2\frac{2x}{x^2-4}-\frac{1}{x^2-3x+2}+\frac{x+1}{x^2+x-2} using a step-by-step approach. We factored the denominators, simplified the expression, and combined the fractions. The final answer is x+1x+2+1\boxed{\frac{x+1}{x+2}+1}.

===========================================================

Introduction

In our previous article, we simplified the expression 2xx2−4−1x2−3x+2+x+1x2+x−2\frac{2x}{x^2-4}-\frac{1}{x^2-3x+2}+\frac{x+1}{x^2+x-2} using a step-by-step approach. In this article, we will provide a Q&A guide to help you understand the process and answer any questions you may have.

Q: What is the first step in simplifying the expression?

A: The first step in simplifying the expression is to factor the denominators. This involves breaking down the denominators into their prime factors.

Q: How do I factor the denominators?

A: To factor the denominators, you can use the following steps:

  1. Look for two numbers that multiply to give the constant term (in this case, -4).
  2. Look for two numbers that add to give the coefficient of the middle term (in this case, 0).
  3. Write the two numbers as factors of the expression.

For example, to factor the denominator x2−4x^2-4, you can write it as (x−2)(x+2)(x-2)(x+2).

Q: What is the next step in simplifying the expression?

A: The next step in simplifying the expression is to simplify the expression by combining the fractions.

Q: How do I combine the fractions?

A: To combine the fractions, you can follow these steps:

  1. Find the least common multiple (LCM) of the denominators.
  2. Rewrite each fraction with the LCM as the denominator.
  3. Add or subtract the numerators.

For example, to combine the fractions 2xx2−4\frac{2x}{x^2-4} and 1x2−3x+2\frac{1}{x^2-3x+2}, you can rewrite them as 2x(x−2)(x+2)\frac{2x}{(x-2)(x+2)} and 1(x−1)(x−2)\frac{1}{(x-1)(x-2)}, respectively.

Q: What is the final step in simplifying the expression?

A: The final step in simplifying the expression is to simplify the resulting expression.

Q: How do I simplify the resulting expression?

A: To simplify the resulting expression, you can follow these steps:

  1. Look for any common factors in the numerator and denominator.
  2. Cancel out any common factors.
  3. Simplify the resulting expression.

For example, to simplify the expression x+1x+2+1\frac{x+1}{x+2}+1, you can rewrite it as x+1x+2+x+2x+2\frac{x+1}{x+2}+\frac{x+2}{x+2}, and then cancel out the common factor (x+2)(x+2).

Q: What is the final answer?

A: The final answer is x+1x+2+1\boxed{\frac{x+1}{x+2}+1}.

Q: Can I use a calculator to simplify the expression?

A: Yes, you can use a calculator to simplify the expression. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Q: Can I use a computer program to simplify the expression?

A: Yes, you can use a computer program such as Sympy to simplify the expression. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Q: What if I get stuck on a step?

A: If you get stuck on a step, don't worry! You can always ask for help or try a different approach. Remember, practice makes perfect, so don't be afraid to try again.

Conclusion

In this article, we provided a Q&A guide to help you understand the process of simplifying the expression 2xx2−4−1x2−3x+2+x+1x2+x−2\frac{2x}{x^2-4}-\frac{1}{x^2-3x+2}+\frac{x+1}{x^2+x-2}. We covered the steps involved in simplifying the expression, including factoring the denominators, combining the fractions, and simplifying the resulting expression. We also answered some common questions that you may have.