Give The Excluded Value(s) For The Following Fraction:${ \frac{y+5}{y^2+4y-32} }$A. { -5, -4$}$ B. ${ 8, -4\$} C. { -8, 4$}$

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Understanding Excluded Values

In mathematics, excluded values refer to the values of a variable that make a mathematical expression undefined or invalid. When dealing with fractions, excluded values occur when the denominator of the fraction is equal to zero. In this article, we will focus on finding the excluded values for a given fraction.

The Given Fraction

The given fraction is y+5y2+4y−32\frac{y+5}{y^2+4y-32}. To find the excluded values, we need to determine the values of yy that make the denominator equal to zero.

Finding the Excluded Values

To find the excluded values, we need to solve the equation y2+4y−32=0y^2+4y-32=0. This is a quadratic equation, and we can solve it using the quadratic formula or by factoring.

Factoring the Quadratic Equation

We can factor the quadratic equation as follows:

y2+4y−32=0y^2+4y-32=0

(y+8)(y−4)=0(y+8)(y-4)=0

This tells us that either (y+8)=0(y+8)=0 or (y−4)=0(y-4)=0.

Solving for y

Solving for yy, we get:

y+8=0⇒y=−8y+8=0 \Rightarrow y=-8

y−4=0⇒y=4y-4=0 \Rightarrow y=4

Therefore, the excluded values are y=−8y=-8 and y=4y=4.

Conclusion

In conclusion, the excluded values for the given fraction y+5y2+4y−32\frac{y+5}{y^2+4y-32} are y=−8y=-8 and y=4y=4. These values make the denominator equal to zero, and therefore, the fraction is undefined at these points.

Comparison with Answer Choices

Now, let's compare our answer with the given answer choices:

A. −5,−4-5, -4 B. 8,−48, -4 C. −8,4-8, 4

Our answer matches with option C. −8,4-8, 4.

Why is this Important?

Understanding excluded values is crucial in mathematics, especially when dealing with fractions. It helps us to identify the values of a variable that make a mathematical expression undefined or invalid. This knowledge is essential in various mathematical applications, such as algebra, calculus, and statistics.

Real-World Applications

Excluded values have numerous real-world applications. For example, in physics, excluded values can help us to determine the values of a variable that make a physical system undefined or invalid. In engineering, excluded values can help us to design systems that avoid certain values of a variable that could lead to system failure.

Common Mistakes to Avoid

When finding excluded values, it's essential to avoid common mistakes such as:

  • Not considering all possible values of a variable
  • Not checking for extraneous solutions
  • Not using the correct method to solve the equation

Tips and Tricks

Here are some tips and tricks to help you find excluded values:

  • Always check the denominator of a fraction to see if it's equal to zero
  • Use the quadratic formula or factoring to solve quadratic equations
  • Be careful when solving equations, and make sure to check for extraneous solutions

Conclusion

Q: What are excluded values?

A: Excluded values are the values of a variable that make a mathematical expression undefined or invalid. In the context of fractions, excluded values occur when the denominator of the fraction is equal to zero.

Q: Why are excluded values important?

A: Excluded values are important because they help us to identify the values of a variable that make a mathematical expression undefined or invalid. This knowledge is essential in various mathematical applications, such as algebra, calculus, and statistics.

Q: How do I find excluded values?

A: To find excluded values, you need to determine the values of a variable that make the denominator of a fraction equal to zero. You can do this by solving the equation that results from setting the denominator equal to zero.

Q: What is the difference between excluded values and asymptotes?

A: Excluded values and asymptotes are related concepts, but they are not the same thing. Excluded values occur when the denominator of a fraction is equal to zero, while asymptotes occur when the numerator and denominator of a fraction have the same degree and the leading coefficients are not equal.

Q: Can excluded values be positive or negative?

A: Yes, excluded values can be positive or negative. For example, if the denominator of a fraction is x2−4x+4x^2-4x+4, the excluded value is x=2x=2, which is a positive value.

Q: Can excluded values be complex numbers?

A: Yes, excluded values can be complex numbers. For example, if the denominator of a fraction is x2+1x^2+1, the excluded value is x=ix=i, which is a complex number.

Q: How do I know if a value is an excluded value or not?

A: To determine if a value is an excluded value or not, you need to check if the denominator of the fraction is equal to zero at that value. If the denominator is equal to zero, then the value is an excluded value.

Q: Can I have multiple excluded values for a single fraction?

A: Yes, you can have multiple excluded values for a single fraction. For example, if the denominator of a fraction is (x−2)(x−3)(x-2)(x-3), the excluded values are x=2x=2 and x=3x=3.

Q: How do I graph excluded values on a number line?

A: To graph excluded values on a number line, you need to mark the excluded values with a dot or a circle. This will help you to visualize the excluded values and understand how they relate to the rest of the number line.

Q: Can I have excluded values that are not integers?

A: Yes, you can have excluded values that are not integers. For example, if the denominator of a fraction is x2−4x+3x^2-4x+3, the excluded value is x=32x=\frac{3}{2}, which is a non-integer value.

Q: How do I use excluded values in real-world applications?

A: Excluded values have numerous real-world applications, such as:

  • Physics: Excluded values can help you to determine the values of a variable that make a physical system undefined or invalid.
  • Engineering: Excluded values can help you to design systems that avoid certain values of a variable that could lead to system failure.
  • Statistics: Excluded values can help you to identify the values of a variable that make a statistical model undefined or invalid.

Q: What are some common mistakes to avoid when working with excluded values?

A: Some common mistakes to avoid when working with excluded values include:

  • Not considering all possible values of a variable
  • Not checking for extraneous solutions
  • Not using the correct method to solve the equation

Q: How do I practice working with excluded values?

A: To practice working with excluded values, you can try the following:

  • Work on problems that involve finding excluded values
  • Practice solving equations that result from setting the denominator equal to zero
  • Use online resources or math software to help you visualize and understand excluded values.