Which Equation Can Be Rewritten As $x + 4 = X^2$? Assume $x \ \textgreater \ 0$.A. $\sqrt{x} + 2 = X$B. $\sqrt{x + 2} = X$C. $\sqrt{x + 4} = X$D. $\sqrt{x^2 + 16} = X$

Introduction

In this article, we will explore the equation $x + 4 = x^2$ and determine which of the given options can be rewritten in this form. We will assume that $x \ \textgreater \ 0$, which means that the variable x is a positive number.

Understanding the Equation

The given equation is a quadratic equation, which is a polynomial equation of degree two. It can be written in the standard form as $ax^2 + bx + c = 0$, where a, b, and c are constants. In this case, the equation is $x^2 - x - 4 = 0$.

Option A: $\sqrt{x} + 2 = x$

Let's start by analyzing the first option, $\sqrt{x} + 2 = x$. To determine if this equation can be rewritten as $x + 4 = x^2$, we need to isolate the variable x.

First, we can subtract 2 from both sides of the equation to get $\sqrt{x} = x - 2$.

Next, we can square both sides of the equation to get $x = (x - 2)^2$.

Expanding the right-hand side of the equation, we get $x = x^2 - 4x + 4$.

Rearranging the terms, we get $x^2 - 5x + 4 = 0$.

This is a quadratic equation, but it is not in the form $x + 4 = x^2$. Therefore, option A is not the correct answer.

Option B: $\sqrt{x + 2} = x$

Now, let's analyze the second option, $\sqrt{x + 2} = x$. To determine if this equation can be rewritten as $x + 4 = x^2$, we need to isolate the variable x.

First, we can square both sides of the equation to get $x + 2 = x^2$.

This is the same as the original equation, $x + 4 = x^2$. Therefore, option B is the correct answer.

Option C: $\sqrt{x + 4} = x$

Now, let's analyze the third option, $\sqrt{x + 4} = x$. To determine if this equation can be rewritten as $x + 4 = x^2$, we need to isolate the variable x.

First, we can square both sides of the equation to get $x + 4 = x^2$.

This is the same as the original equation, $x + 4 = x^2$. However, we need to check if this equation is consistent with the assumption that $x \ \textgreater \ 0$.

Squaring both sides of the equation, we get $x^2 + 8x + 16 = x^4$.

Rearranging the terms, we get $x^4 - x^2 - 8x - 16 = 0$.

This is a quartic equation, which is a polynomial equation of degree four. It is not a quadratic equation, and it is not in the form $x + 4 = x^2$. Therefore, option C is not the correct answer.

Option D: $\sqrt{x^2 + 16} = x$

Now, let's analyze the fourth option, $\sqrt{x^2 + 16} = x$. To determine if this equation can be rewritten as $x + 4 = x^2$, we need to isolate the variable x.

First, we can square both sides of the equation to get $x^2 + 16 = x^2$.

This is a true statement, but it is not in the form $x + 4 = x^2$. Therefore, option D is not the correct answer.

Conclusion

In conclusion, the correct answer is option B, $\sqrt{x + 2} = x$. This equation can be rewritten as $x + 4 = x^2$, and it is consistent with the assumption that $x \ \textgreater \ 0$.

Final Answer

The final answer is B.