Which Choice Is The Conjugate Of The Expression Below When $x \geq -4$?$5-\sqrt{x+4}$A. \$5-\sqrt{x-4}$[/tex\] B. $5+\sqrt{x+4}$ C. $5+\sqrt{x-4}$ D. \$5-\sqrt{x+4}$[/tex\]
**Conjugate of an Expression: A Comprehensive Guide** =====================================================
What is a Conjugate in Mathematics?
In mathematics, a conjugate is a pair of expressions that have the same value when one is added to the other. Conjugates are commonly used in algebra, particularly when dealing with square roots and other irrational numbers. In this article, we will explore the concept of conjugates and how to find the conjugate of a given expression.
What is the Conjugate of an Expression?
The conjugate of an expression is another expression that, when added to the original expression, results in a rational number. In other words, the conjugate of an expression is a way to eliminate the radical sign by adding its opposite.
How to Find the Conjugate of an Expression?
To find the conjugate of an expression, we need to follow these steps:
- Identify the radical sign: The first step is to identify the radical sign in the expression. In the given expression $x \geq -4$, the radical sign is $\sqrt{x+4}$.
- Change the sign: The next step is to change the sign of the expression inside the radical sign. In this case, we change the sign of $x+4$ to $-(x+4)$.
- Simplify the expression: Finally, we simplify the expression by combining like terms. In this case, $-(x+4)$ simplifies to $-x-4$.
Finding the Conjugate of the Given Expression
Now that we have understood the concept of conjugates and how to find them, let's apply this knowledge to the given expression $5-\sqrt{x+4}$.
To find the conjugate of this expression, we need to follow the steps outlined above:
- Identify the radical sign: The radical sign in the expression is $\sqrt{x+4}$.
- Change the sign: We change the sign of $x+4$ to $-(x+4)$.
- Simplify the expression: We simplify the expression by combining like terms. In this case, $-(x+4)$ simplifies to $-x-4$.
Therefore, the conjugate of the expression $5-\sqrt{x+4}$ is $5-\sqrt{x+4}$.
Answer to the Question
The correct answer to the question is:
- D. $5-\sqrt{x+4}$
Conclusion
In conclusion, the conjugate of an expression is another expression that, when added to the original expression, results in a rational number. To find the conjugate of an expression, we need to follow the steps outlined above: identify the radical sign, change the sign, and simplify the expression. By applying this knowledge to the given expression $5-\sqrt{x+4}$, we have found that the conjugate of this expression is also $5-\sqrt{x+4}$.
Frequently Asked Questions (FAQs)
Q: What is a conjugate in mathematics?
A: A conjugate in mathematics is a pair of expressions that have the same value when one is added to the other.
Q: How do I find the conjugate of an expression?
A: To find the conjugate of an expression, you need to follow these steps: identify the radical sign, change the sign, and simplify the expression.
Q: What is the conjugate of the expression $5-\sqrt{x+4}$?
A: The conjugate of the expression $5-\sqrt{x+4}$ is also $5-\sqrt{x+4}$.
Q: Why is it important to find the conjugate of an expression?
A: Finding the conjugate of an expression is important because it allows us to eliminate the radical sign and simplify the expression.
Q: Can you give an example of a conjugate pair?
A: Yes, an example of a conjugate pair is $\sqrt2}$ and $-\sqrt{2}$. When added together, they result in a rational number + (-\sqrt{2}) = 0$.
Q: How do I know if an expression has a conjugate?
A: An expression has a conjugate if it contains a radical sign. In this case, you can find the conjugate by following the steps outlined above.