What Are The Excluded Values For $\frac{-7}{x^2 - 2x - 15}$?A. -3 And 5 B. 3 And -5 C. 0 And 15

Understanding Excluded Values in Rational Expressions

When dealing with rational expressions, it's essential to understand the concept of excluded values. Excluded values are the values of the variable that make the denominator of the rational expression equal to zero. In other words, they are the values that would result in a division by zero, which is undefined in mathematics.

The Rational Expression in Question

The rational expression in question is $\frac{-7}{x^2 - 2x - 15}$. To find the excluded values, we need to find the values of $x$ that make the denominator equal to zero.

Finding the Excluded Values

To find the excluded values, we need to solve the equation $x^2 - 2x - 15 = 0$. 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:

$x^2 - 2x - 15 = (x - 5)(x + 3) = 0$

This tells us that either $(x - 5) = 0$ or $(x + 3) = 0$.

Solving for x

Solving for $x$, we get:

$x - 5 = 0 \Rightarrow x = 5$

$x + 3 = 0 \Rightarrow x = -3$

Therefore, the excluded values are $x = 5$ and $x = -3$.

Conclusion

In conclusion, the excluded values for the rational expression $\frac{-7}{x^2 - 2x - 15}$ are $x = 5$ and $x = -3$. These values make the denominator equal to zero, resulting in a division by zero, which is undefined in mathematics.

Answer

The correct answer is A. -3 and 5.

Additional Tips and Examples

• When dealing with rational expressions, it's essential to identify the excluded values to avoid division by zero.
• Excluded values can be found by setting the denominator equal to zero and solving for the variable.
• In this example, we used factoring to solve the quadratic equation. However, we can also use the quadratic formula to solve it.

Quadratic Formula

The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$, $b = -2$, and $c = -15$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-15)}}{2(1)}$

Simplifying, we get:

$x = \frac{2 \pm \sqrt{4 + 60}}{2}$

$x = \frac{2 \pm \sqrt{64}}{2}$

$x = \frac{2 \pm 8}{2}$

This gives us two possible values for $x$:

$x = \frac{2 + 8}{2} = 5$

$x = \frac{2 - 8}{2} = -3$

Therefore, the excluded values are still $x = 5$ and $x = -3$.

Real-World Applications

Excluded values have real-world applications in various fields, such as engineering, physics, and economics. For example, in engineering, excluded values can be used to determine the stability of a system. In physics, excluded values can be used to determine the energy levels of a particle. In economics, excluded values can be used to determine the optimal price of a product.

Conclusion

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Frequently Asked Questions

Q: What are excluded values in rational expressions?

A: Excluded values are the values of the variable that make the denominator of the rational expression equal to zero. In other words, they are the values that would result in a division by zero, which is undefined in mathematics.

Q: How do I find the excluded values in a rational expression?

A: To find the excluded values, you need to solve the equation that makes the denominator equal to zero. This can be done using factoring, the quadratic formula, or other methods.

Q: What happens if I don't consider the excluded values in a rational expression?

A: If you don't consider the excluded values, you may end up with a division by zero, which is undefined in mathematics. This can lead to incorrect results and errors in your calculations.

Q: Can I have multiple excluded values in a rational expression?

A: Yes, it's possible to have multiple excluded values in a rational expression. This occurs when the denominator can be factored into multiple linear factors, each of which can be set equal to zero.

Q: How do I determine the excluded values in a rational expression with a quadratic denominator?

A: To determine the excluded values in a rational expression with a quadratic denominator, you need to solve the quadratic equation that makes the denominator equal to zero. This can be done using factoring, the quadratic formula, or other methods.

Q: Can I use the quadratic formula to find the excluded values in a rational expression?

A: Yes, you can use the quadratic formula to find the excluded values in a rational expression. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: What are some real-world applications of excluded values in rational expressions?

A: Excluded values have real-world applications in various fields, such as engineering, physics, and economics. For example, in engineering, excluded values can be used to determine the stability of a system. In physics, excluded values can be used to determine the energy levels of a particle. In economics, excluded values can be used to determine the optimal price of a product.

Q: Can I have a rational expression with no excluded values?

A: Yes, it's possible to have a rational expression with no excluded values. This occurs when the denominator is a constant or a linear expression that is never equal to zero.

Q: How do I simplify a rational expression with excluded values?

A: To simplify a rational expression with excluded values, you need to cancel out any common factors between the numerator and denominator. However, you should not cancel out any factors that would result in a division by zero.

Q: Can I have a rational expression with multiple excluded values and multiple factors in the numerator?

A: Yes, it's possible to have a rational expression with multiple excluded values and multiple factors in the numerator. In this case, you need to cancel out any common factors between the numerator and denominator, while avoiding any factors that would result in a division by zero.

Q: How do I determine the excluded values in a rational expression with a polynomial denominator?

A: To determine the excluded values in a rational expression with a polynomial denominator, you need to solve the equation that makes the denominator equal to zero. This can be done using factoring, the quadratic formula, or other methods.

Q: Can I use technology to find the excluded values in a rational expression?

A: Yes, you can use technology, such as graphing calculators or computer algebra systems, to find the excluded values in a rational expression. These tools can help you solve the equation that makes the denominator equal to zero and find the excluded values.

Q: How do I apply the concept of excluded values in real-world problems?

A: To apply the concept of excluded values in real-world problems, you need to identify the excluded values in the rational expression and use them to determine the solution to the problem. This may involve canceling out common factors, avoiding division by zero, and using the excluded values to determine the optimal solution.

Conclusion

In conclusion, excluded values are an essential concept in mathematics, particularly in rational expressions. By understanding the concept of excluded values, you can avoid division by zero and ensure that your calculations are accurate. The Q&A section above provides answers to frequently asked questions about excluded values in rational expressions.