Identify The Expression That Is Equal To X 2 + X − X \sqrt{x^2+x}-x X 2 + X − X :A. X 2 + X − X = ( X 2 + X − X ) ⋅ X 2 + X + X X 2 + X + X \sqrt{x^2+x}-x=\left(\sqrt{x^2+x}-x\right) \cdot \frac{\sqrt{x^2+x}+x}{\sqrt{x^2+x}+x} X 2 + X − X = ( X 2 + X − X ) ⋅ X 2 + X + X X 2 + X + X B. $\sqrt{x 2+x}-x=\left(\sqrt{x 2+x}-x\right) \cdot

Mar 8, 2025 by ADMIN 348 views

**Simplifying Radical Expressions: A Step-by-Step Guide**

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression $\sqrt{x^2+x}-x$ . We will examine two possible solutions and determine which one is correct.

Understanding the Given Expression

The given expression is $\sqrt{x^2+x}-x$ . This expression involves a square root and a variable $x$ . To simplify this expression, we need to understand the properties of square roots and how they interact with variables.

Properties of Square Roots

Square roots have several important properties that we need to understand when simplifying radical expressions. These properties include:

Non-Negativity: The square root of a number is always non-negative, i.e., $\sqrt{x} \geq 0$ for all $x \geq 0$ .
Monotonicity: The square root function is monotonically increasing, i.e., $\sqrt{x} < \sqrt{y}$ if $x < y$ .
Idempotence: The square root function is idempotent, i.e., $\sqrt{\sqrt{x}} = x$ for all $x \geq 0$ .

Simplifying the Given Expression

To simplify the given expression, we can use the properties of square roots mentioned above. Let's examine the two possible solutions:

Option A: Multiplying by the Conjugate

The first option is to multiply the given expression by the conjugate of the numerator:

$\sqrt{x^2+x}-x=\left(\sqrt{x^2+x}-x\right) \cdot \frac{\sqrt{x^2+x}+x}{\sqrt{x^2+x}+x}$

This option involves multiplying the numerator and denominator by the conjugate of the numerator, which is $\sqrt{x^2+x}+x$ . The conjugate of a binomial expression $a+b$ is $a-b$ , so in this case, the conjugate of $\sqrt{x^2+x}$ is $-\sqrt{x^2+x}$ .

Option B: Multiplying by the Conjugate of the Denominator

The second option is to multiply the given expression by the conjugate of the denominator:

$\sqrt{x^2+x}-x=\left(\sqrt{x^2+x}-x\right) \cdot \frac{\sqrt{x^2+x}-x}{\sqrt{x^2+x}-x}$

This option involves multiplying the numerator and denominator by the conjugate of the denominator, which is $\sqrt{x^2+x}-x$ . The conjugate of a binomial expression $a+b$ is $a-b$ , so in this case, the conjugate of $\sqrt{x^2+x}+x$ is $\sqrt{x^2+x}-x$ .

Evaluating the Options

Now that we have examined the two possible solutions, let's evaluate them:

Option A: Multiplying by the Conjugate

When we multiply the given expression by the conjugate of the numerator, we get:

$\left(\sqrt{x^2+x}-x\right) \cdot \frac{\sqrt{x^2+x}+x}{\sqrt{x^2+x}+x} = \frac{x^2+x-x^2}{\sqrt{x^2+x}+x} = \frac{x}{\sqrt{x^2+x}+x}$

This expression is not equal to the original expression $\sqrt{x^2+x}-x$ , so option A is not correct.

Option B: Multiplying by the Conjugate of the Denominator

When we multiply the given expression by the conjugate of the denominator, we get:

$\left(\sqrt{x^2+x}-x\right) \cdot \frac{\sqrt{x^2+x}-x}{\sqrt{x^2+x}-x} = \frac{x^2+x-x^2}{\sqrt{x^2+x}-x} = \frac{x}{\sqrt{x^2+x}-x}$

This expression is not equal to the original expression $\sqrt{x^2+x}-x$ , so option B is not correct.

Conclusion

In this article, we examined two possible solutions for simplifying the given expression $\sqrt{x^2+x}-x$ . We evaluated both options and determined that neither of them is correct. The correct solution is not among the options provided.

Final Answer

Introduction

In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression $\sqrt{x^2+x}-x$ . We examined two possible solutions and determined that neither of them is correct. In this article, we will provide a Q&A guide to help you better understand the concept of simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that involves a square root or other root. It is a mathematical expression that contains a root symbol, such as $\sqrt{x}$ or $\sqrt[3]{x}$ .

Q: What are the properties of square roots?

A: The properties of square roots include:

Non-Negativity: The square root of a number is always non-negative, i.e., $\sqrt{x} \geq 0$ for all $x \geq 0$ .
Monotonicity: The square root function is monotonically increasing, i.e., $\sqrt{x} < \sqrt{y}$ if $x < y$ .
Idempotence: The square root function is idempotent, i.e., $\sqrt{\sqrt{x}} = x$ for all $x \geq 0$ .

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can use the following steps:

Rationalize the denominator: If the expression has a denominator that contains a square root, you can multiply the numerator and denominator by the conjugate of the denominator to rationalize the denominator.
Simplify the numerator: Once you have rationalized the denominator, you can simplify the numerator by combining like terms.
Simplify the expression: Finally, you can simplify the expression by combining like terms and eliminating any unnecessary radicals.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression $a+b$ is $a-b$ . For example, the conjugate of $x+y$ is $x-y$ .

Q: How do I multiply a binomial expression by its conjugate?

A: To multiply a binomial expression by its conjugate, you can use the following steps:

Multiply the numerator: Multiply the numerator of the binomial expression by the conjugate of the denominator.
Multiply the denominator: Multiply the denominator of the binomial expression by the conjugate of the denominator.
Simplify the expression: Finally, you can simplify the expression by combining like terms and eliminating any unnecessary radicals.

Q: What is the difference between a rational expression and a radical expression?

A: A rational expression is an expression that contains a fraction, such as $\frac{x}{y}$ . A radical expression is an expression that contains a root symbol, such as $\sqrt{x}$ .

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you can use the following steps:

Factor the numerator and denominator: Factor the numerator and denominator of the expression to simplify it.
Cancel out common factors: Cancel out any common factors between the numerator and denominator.
Simplify the expression: Finally, you can simplify the expression by combining like terms and eliminating any unnecessary fractions.

Conclusion

In this article, we provided a Q&A guide to help you better understand the concept of simplifying radical expressions. We covered topics such as the properties of square roots, how to simplify a radical expression, and how to multiply a binomial expression by its conjugate. We also discussed the difference between a rational expression and a radical expression, and how to simplify a rational expression.