The Expression $\frac{x^2-9}{x^2+6x+8}$ Is Equal To Zero When $x =$ A. 3 And -3B. 9 And 9C. 4 And 2D. -4 And -2

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Introduction

To find the values of $x$ for which the expression $\frac{x^2-9}{x^2+6x+8}$ is equal to zero, we need to find the values of $x$ that make the numerator of the expression equal to zero. This is because a fraction is equal to zero only when its numerator is equal to zero, and its denominator is not equal to zero.

Step 1: Factor the Numerator

The numerator of the expression is $x^2-9$. We can factor this expression as $(x-3)(x+3)$.

Step 2: Set the Numerator Equal to Zero

To find the values of $x$ that make the numerator equal to zero, we set the factored form of the numerator equal to zero: $(x-3)(x+3) = 0$.

Step 3: Solve for $x$

We can solve for $x$ by setting each factor equal to zero and solving for $x$. This gives us two possible values for $x$: $x = 3$ and $x = -3$.

Step 4: Check the Denominator

Before we can conclude that the expression is equal to zero when $x = 3$ and $x = -3$, we need to check that the denominator is not equal to zero at these values of $x$. The denominator is $x^2+6x+8$, which can be factored as $(x+4)(x+2)$. We can see that the denominator is not equal to zero when $x = 3$ and $x = -3$, since $(3+4)(3+2) = 7 \cdot 5 \neq 0$ and $(-3+4)(-3+2) = 1 \cdot -1 \neq 0$.

Conclusion

Therefore, the expression $\frac{x^2-9}{x^2+6x+8}$ is equal to zero when $x = 3$ and $x = -3$.

Answer Options

A. 3 and -3 B. 9 and 9 C. 4 and 2 D. -4 and -2

Final Answer

The final answer is A. 3 and -3.

Explanation

The expression $\frac{x^2-9}{x^2+6x+8}$ is equal to zero when the numerator is equal to zero and the denominator is not equal to zero. The numerator is equal to zero when $x = 3$ and $x = -3$, and the denominator is not equal to zero at these values of $x$. Therefore, the expression is equal to zero when $x = 3$ and $x = -3$.

Additional Information

It's worth noting that the expression $\frac{x^2-9}{x^2+6x+8}$ is a rational expression, and rational expressions can be equal to zero only when the numerator is equal to zero and the denominator is not equal to zero. This is a fundamental property of rational expressions, and it's essential to keep it in mind when working with rational expressions.

Example

To illustrate this concept, let's consider the expression $\frac{x^2-4}{x^2+2x+1}$. We can factor the numerator as $(x-2)(x+2)$, and the denominator as $(x+1)^2$. We can see that the numerator is equal to zero when $x = 2$ and $x = -2$, but the denominator is equal to zero when $x = -1$. Therefore, the expression is not equal to zero when $x = 2$ and $x = -2$, since the denominator is equal to zero at these values of $x$.

Conclusion

In conclusion, the expression $\frac{x^2-9}{x^2+6x+8}$ is equal to zero when $x = 3$ and $x = -3$, since the numerator is equal to zero at these values of $x$ and the denominator is not equal to zero. This is a fundamental property of rational expressions, and it's essential to keep it in mind when working with rational expressions.

Final Answer

The final answer is A. 3 and -3.

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Q&A

Q: What is the numerator of the expression $\frac{x^2-9}{x^2+6x+8}$?

A: The numerator of the expression is $x^2-9$.

Q: How can we factor the numerator $x^2-9$?

A: We can factor the numerator as $(x-3)(x+3)$.

Q: What are the values of $x$ that make the numerator equal to zero?

A: The values of $x$ that make the numerator equal to zero are $x = 3$ and $x = -3$.

Q: What is the denominator of the expression $\frac{x^2-9}{x^2+6x+8}$?

A: The denominator of the expression is $x^2+6x+8$.

Q: How can we factor the denominator $x^2+6x+8$?

A: We can factor the denominator as $(x+4)(x+2)$.

Q: What are the values of $x$ that make the denominator equal to zero?

A: The values of $x$ that make the denominator equal to zero are $x = -4$ and $x = -2$.

Q: Why is it essential to check the denominator when finding the values of $x$ that make the expression equal to zero?

A: It's essential to check the denominator because the expression is only equal to zero when the numerator is equal to zero and the denominator is not equal to zero.

Q: What is the final answer to the problem?

A: The final answer is A. 3 and -3.

Q: Can you provide an example of a rational expression that is not equal to zero when the numerator is equal to zero?

A: Yes, consider the expression $\frac{x^2-4}{x^2+2x+1}$. We can factor the numerator as $(x-2)(x+2)$, and the denominator as $(x+1)^2$. We can see that the numerator is equal to zero when $x = 2$ and $x = -2$, but the denominator is equal to zero when $x = -1$. Therefore, the expression is not equal to zero when $x = 2$ and $x = -2$, since the denominator is equal to zero at these values of $x$.

Q: What is the significance of rational expressions in mathematics?

A: Rational expressions are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, calculus, and engineering. They are used to model real-world problems, and they provide a powerful tool for solving equations and inequalities.

Q: Can you provide a real-world example of a rational expression?

A: Yes, consider the problem of finding the cost of a product that is discounted by a certain percentage. Let's say the original price of the product is $100, and it's discounted by 20%. The cost of the product can be represented by the rational expression $\frac{80}{100}$. This expression represents the cost of the product as a fraction of its original price.

Q: What is the final answer to the problem?

A: The final answer is A. 3 and -3.

Conclusion

In conclusion, the expression $\frac{x^2-9}{x^2+6x+8}$ is equal to zero when $x = 3$ and $x = -3$, since the numerator is equal to zero at these values of $x$ and the denominator is not equal to zero. This is a fundamental property of rational expressions, and it's essential to keep it in mind when working with rational expressions.

Final Answer

The final answer is A. 3 and -3.