Select All The Correct Answers.For Which Values Is This Expression Undefined? X − 1 X 2 − 2 X − 3 + 5 2 X 2 + 2 X \frac{x-1}{x^2-2x-3}+\frac{5}{2x^2+2x} X 2 − 2 X − 3 X − 1 ​ + 2 X 2 + 2 X 5 ​ A. X = 0 X=0 X = 0 B. X = − 1 X=-1 X = − 1 C. X = 1 X=1 X = 1 D. X = 3 X=3 X = 3 E. X = − 2 X=-2 X = − 2 F. X = 5 X=5 X = 5

Introduction

In mathematics, expressions involving fractions can be undefined for certain values of the variable. This is due to the presence of denominators that become zero, leading to division by zero, which is undefined in mathematics. In this article, we will analyze the given expression $\frac{x-1}{x^2-2x-3}+\frac{5}{2x^2+2x}$ and determine the values of $x$ for which it is undefined.

Understanding the Expression

The given expression consists of two fractions added together. To determine the values of $x$ for which the expression is undefined, we need to find the values of $x$ that make the denominators of each fraction equal to zero.

Denominator of the First Fraction

The denominator of the first fraction is $x^2-2x-3$. To find the values of $x$ that make this expression equal to zero, we can factorize it as follows:

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

Setting each factor equal to zero, we get:

$$(x-3) = 0 \Rightarrow x = 3$$

$$(x+1) = 0 \Rightarrow x = -1$$

Therefore, the values of $x$ that make the denominator of the first fraction equal to zero are $x = 3$ and $x = -1$.

Denominator of the Second Fraction

The denominator of the second fraction is $2x^2+2x$. To find the values of $x$ that make this expression equal to zero, we can factorize it as follows:

$$2x^2+2x = 2x(x+1)$$

Setting each factor equal to zero, we get:

$$2x = 0 \Rightarrow x = 0$$

$$(x+1) = 0 \Rightarrow x = -1$$

Therefore, the values of $x$ that make the denominator of the second fraction equal to zero are $x = 0$ and $x = -1$.

Conclusion

In conclusion, the expression $\frac{x-1}{x^2-2x-3}+\frac{5}{2x^2+2x}$ is undefined for the following values of $x$:

• $x = 3$ (denominator of the first fraction)
• $x = -1$ (denominator of the first fraction and the second fraction)
• $x = 0$ (denominator of the second fraction)

Therefore, the correct answers are:

• A. $x=0$
• B. $x=-1$
• C. $x=1$ (not a correct answer)
• D. $x=3$
• E. $x=-2$ (not a correct answer)
• F. $x=5$ (not a correct answer)

Final Answer

Q&A: Understanding Undefined Expression Values

In the previous article, we analyzed the expression $\frac{x-1}{x^2-2x-3}+\frac{5}{2x^2+2x}$ and determined the values of $x$ for which it is undefined. In this article, we will answer some frequently asked questions related to undefined expression values.

Q: What is an undefined expression?

A: An undefined expression is an expression that cannot be evaluated to a specific value due to the presence of a denominator that becomes zero. In other words, an expression is undefined when it is divided by zero.

Q: Why is division by zero undefined?

A: Division by zero is undefined because it leads to a contradiction. For example, if we have the expression $\frac{1}{0}$, it would imply that $1$ is equal to $0$, which is not true. Therefore, division by zero is not allowed in mathematics.

Q: How do I determine the values of $x$ for which an expression is undefined?

A: To determine the values of $x$ for which an expression is undefined, you need to find the values of $x$ that make the denominators of each fraction equal to zero. You can do this by factorizing the denominators and setting each factor equal to zero.

Q: What are some common mistakes to avoid when determining undefined expression values?

A: Some common mistakes to avoid when determining undefined expression values include:

• Not considering all possible values of $x$ that make the denominators equal to zero.
• Not factorizing the denominators correctly.
• Not setting each factor equal to zero to find the values of $x$.

Q: Can I simplify an expression that is undefined?

A: No, you cannot simplify an expression that is undefined. An undefined expression is not a valid mathematical expression, and it cannot be simplified.

Q: How do I handle undefined expression values in real-world applications?

A: In real-world applications, undefined expression values can be handled in several ways, including:

• Avoiding the values of $x$ that make the expression undefined.
• Using alternative expressions that are defined for all values of $x$.
• Using numerical methods to approximate the value of the expression.

Q: Can I use undefined expression values in calculus?

A: No, you cannot use undefined expression values in calculus. Calculus requires the use of defined expressions, and undefined expression values are not valid in calculus.

Conclusion

In conclusion, undefined expression values are an important concept in mathematics that can be challenging to understand. By following the steps outlined in this article, you can determine the values of $x$ for which an expression is undefined and avoid common mistakes. Remember that undefined expression values are not valid mathematical expressions and cannot be simplified.

Final Answer

The final answer is that undefined expression values are an important concept in mathematics that require careful analysis and handling.