Which Expression Is Equivalent To 2 X 2 + 2 X − 4 2 X 2 − 4 X + 2 \frac{2x^2 + 2x - 4}{2x^2 - 4x + 2} 2 X 2 − 4 X + 2 2 X 2 + 2 X − 4 ​ ?A. X + 2 X − 1 \frac{x+2}{x-1} X − 1 X + 2 ​ B. X + 2 X+2 X + 2 C. − 2 -2 − 2 D. X + 2 X − 2 \frac{x+2}{x-2} X − 2 X + 2 ​

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Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying rational expressions, with a focus on the given expression 2x2+2x−42x2−4x+2\frac{2x^2 + 2x - 4}{2x^2 - 4x + 2}. We will examine the different options provided and determine which one is equivalent to the given expression.

Understanding Rational Expressions

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Simplifying a rational expression involves factoring the numerator and denominator, canceling out any common factors, and then reducing the resulting fraction to its simplest form.

Factoring the Numerator and Denominator

To simplify the given expression, we need to factor the numerator and denominator. The numerator can be factored as follows:

2x2+2x−4=2(x2+x−2)2x^2 + 2x - 4 = 2(x^2 + x - 2)

We can further factor the quadratic expression inside the parentheses:

x2+x−2=(x+2)(x−1)x^2 + x - 2 = (x + 2)(x - 1)

So, the numerator can be written as:

2(x+2)(x−1)2(x + 2)(x - 1)

The denominator can be factored as follows:

2x2−4x+2=2(x2−2x+1)2x^2 - 4x + 2 = 2(x^2 - 2x + 1)

We can further factor the quadratic expression inside the parentheses:

x2−2x+1=(x−1)2x^2 - 2x + 1 = (x - 1)^2

So, the denominator can be written as:

2(x−1)22(x - 1)^2

Canceling Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out any common factors. In this case, we have a common factor of (x−1)(x - 1) in both the numerator and denominator. We can cancel out this factor as follows:

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

Evaluating the Options

Now that we have simplified the given expression, we can evaluate the options provided. Let's examine each option in turn:

Option A: x+2x−1\frac{x+2}{x-1}

This option is equivalent to the simplified expression we obtained earlier. Therefore, it is a possible solution.

Option B: x+2x+2

This option is not equivalent to the simplified expression. The simplified expression has a denominator of (x−1)(x - 1), which is not present in this option.

Option C: −2-2

This option is not equivalent to the simplified expression. The simplified expression has a variable in the numerator and denominator, whereas this option is a constant.

Option D: x+2x−2\frac{x+2}{x-2}

This option is not equivalent to the simplified expression. The simplified expression has a denominator of (x−1)(x - 1), whereas this option has a denominator of (x−2)(x - 2).

Conclusion

In conclusion, the correct answer is Option A: x+2x−1\frac{x+2}{x-1}. This option is equivalent to the simplified expression we obtained earlier. The other options are not equivalent to the simplified expression and can be eliminated.

Final Answer

The final answer is x+2x−1\boxed{\frac{x+2}{x-1}}.

Frequently Asked Questions

Q: What is the process of simplifying rational expressions?

A: The process of simplifying rational expressions involves factoring the numerator and denominator, canceling out any common factors, and then reducing the resulting fraction to its simplest form.

Q: How do I factor the numerator and denominator?

A: To factor the numerator and denominator, you need to identify any common factors and factor them out. You can also use the quadratic formula to factor quadratic expressions.

Q: What is the difference between a rational expression and a fraction?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. A fraction is a general term that refers to a ratio of two numbers.

Q: How do I cancel out common factors?

A: To cancel out common factors, you need to identify any common factors in the numerator and denominator and divide them out.

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Introduction

Rational expressions are a fundamental concept in algebra, and understanding them is crucial for solving equations and inequalities. In this article, we will provide a comprehensive Q&A guide to help you master rational expressions.

Q&A Section

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, cancel out any common factors, and then reduce the resulting fraction to its simplest form.

Q: What is the difference between a rational expression and a fraction?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. A fraction is a general term that refers to a ratio of two numbers.

Q: How do I factor the numerator and denominator?

A: To factor the numerator and denominator, you need to identify any common factors and factor them out. You can also use the quadratic formula to factor quadratic expressions.

Q: What is the difference between a rational expression and an algebraic expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. An algebraic expression is a general term that refers to any expression that contains variables and/or constants.

Q: How do I cancel out common factors?

A: To cancel out common factors, you need to identify any common factors in the numerator and denominator and divide them out.

Q: What is the rule for canceling out common factors?

A: The rule for canceling out common factors is that you can only cancel out common factors if they are present in both the numerator and denominator.

Q: How do I reduce a rational expression to its simplest form?

A: To reduce a rational expression to its simplest form, you need to cancel out any common factors and then simplify the resulting fraction.

Q: What is the difference between a rational expression and a polynomial?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. A polynomial is a general term that refers to any expression that contains variables and/or constants and is written in the form of a sum of terms.

Q: How do I add or subtract rational expressions?

A: To add or subtract rational expressions, you need to find a common denominator and then add or subtract the numerators.

Q: What is the rule for adding or subtracting rational expressions?

A: The rule for adding or subtracting rational expressions is that you need to find a common denominator and then add or subtract the numerators.

Q: How do I multiply rational expressions?

A: To multiply rational expressions, you need to multiply the numerators and denominators separately.

Q: What is the rule for multiplying rational expressions?

A: The rule for multiplying rational expressions is that you need to multiply the numerators and denominators separately.

Q: How do I divide rational expressions?

A: To divide rational expressions, you need to invert the second rational expression and then multiply.

Q: What is the rule for dividing rational expressions?

A: The rule for dividing rational expressions is that you need to invert the second rational expression and then multiply.

Conclusion

In conclusion, rational expressions are a fundamental concept in algebra, and understanding them is crucial for solving equations and inequalities. By following the rules and guidelines outlined in this Q&A guide, you can master rational expressions and become proficient in solving equations and inequalities.

Final Tips

  • Always simplify rational expressions before solving equations or inequalities.
  • Use the rules for adding, subtracting, multiplying, and dividing rational expressions to solve equations and inequalities.
  • Practice, practice, practice! The more you practice, the more comfortable you will become with rational expressions.

Frequently Asked Questions

Q: What is the difference between a rational expression and a fraction?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. A fraction is a general term that refers to a ratio of two numbers.

Q: How do I factor the numerator and denominator?

A: To factor the numerator and denominator, you need to identify any common factors and factor them out. You can also use the quadratic formula to factor quadratic expressions.

Q: What is the rule for canceling out common factors?

A: The rule for canceling out common factors is that you can only cancel out common factors if they are present in both the numerator and denominator.

Q: How do I reduce a rational expression to its simplest form?

A: To reduce a rational expression to its simplest form, you need to cancel out any common factors and then simplify the resulting fraction.

Q: What is the difference between a rational expression and a polynomial?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. A polynomial is a general term that refers to any expression that contains variables and/or constants and is written in the form of a sum of terms.

Additional Resources

  • Algebra textbooks and online resources
  • Online tutorials and video lessons
  • Practice problems and worksheets

Conclusion

In conclusion, rational expressions are a fundamental concept in algebra, and understanding them is crucial for solving equations and inequalities. By following the rules and guidelines outlined in this Q&A guide, you can master rational expressions and become proficient in solving equations and inequalities.