Solve For $x$.$\sqrt{x-3}=\sqrt{x}+3$Possible Answers:A. 4 B. No Solution C. 2 D. $-2$

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Understanding the Problem

When solving equations involving square roots, it's essential to consider the domain of the square root function. The expression inside the square root must be non-negative. In this case, we have $\sqrt{x-3}=\sqrt{x}+3$. Our goal is to find the value of $x$ that satisfies this equation.

Isolating the Square Root Terms

To begin solving the equation, we can isolate the square root terms on one side. We'll start by subtracting $\sqrt{x}$ from both sides of the equation:

$$\sqrt{x-3}-\sqrt{x}=3$$

Rationalizing the Denominator

To eliminate the square roots, we can multiply both sides of the equation by the conjugate of the left-hand side. The conjugate of $\sqrt{x-3}-\sqrt{x}$ is $\sqrt{x-3}+\sqrt{x}$. Multiplying both sides by this expression gives:

$$(\sqrt{x-3}-\sqrt{x})(\sqrt{x-3}+\sqrt{x})=3(\sqrt{x-3}+\sqrt{x})$$

Expanding and Simplifying

Expanding the left-hand side of the equation, we get:

$$x-3-x=3(\sqrt{x-3}+\sqrt{x})$$

Simplifying the left-hand side, we have:

$$-3=3(\sqrt{x-3}+\sqrt{x})$$

Dividing by 3

Dividing both sides of the equation by 3 gives:

$$-\frac{3}{3}=\sqrt{x-3}+\sqrt{x}$$

Simplifying the Right-Hand Side

Simplifying the right-hand side of the equation, we get:

$$-1=\sqrt{x-3}+\sqrt{x}$$

Subtracting $\sqrt{x}$ from Both Sides

Subtracting $\sqrt{x}$ from both sides of the equation gives:

$$-1-\sqrt{x}=\sqrt{x-3}$$

Squaring Both Sides

Squaring both sides of the equation, we get:

$$(-1-\sqrt{x})^2=(\sqrt{x-3})^2$$

Expanding the Left-Hand Side

Expanding the left-hand side of the equation, we get:

$$1+2\sqrt{x}+x=x-3$$

Simplifying the Left-Hand Side

Simplifying the left-hand side of the equation, we get:

$$1+2\sqrt{x}+x-x=-(x-3)$$

Simplifying the Right-Hand Side

Simplifying the right-hand side of the equation, we get:

$$1+2\sqrt{x}=-(x-3)$$

Multiplying Both Sides by -1

Multiplying both sides of the equation by -1 gives:

$$-1-2\sqrt{x}=x-3$$

Adding 3 to Both Sides

Adding 3 to both sides of the equation gives:

$$2-2\sqrt{x}=x$$

Subtracting 2 from Both Sides

Subtracting 2 from both sides of the equation gives:

$$-2\sqrt{x}=x-2$$

Dividing by -2

Dividing both sides of the equation by -2 gives:

$$\sqrt{x}=\frac{x-2}{-2}$$

Simplifying the Right-Hand Side

Simplifying the right-hand side of the equation, we get:

$$\sqrt{x}=-\frac{x-2}{2}$$

Squaring Both Sides

Squaring both sides of the equation, we get:

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

Expanding the Right-Hand Side

Expanding the right-hand side of the equation, we get:

$$x=\frac{(x-2)^2}{4}$$

Multiplying Both Sides by 4

Multiplying both sides of the equation by 4 gives:

$$4x=(x-2)^2$$

Expanding the Right-Hand Side

Expanding the right-hand side of the equation, we get:

$$4x=x^2-4x+4$$

Rearranging the Terms

Rearranging the terms of the equation, we get:

$$0=x^2-8x+4$$

Factoring the Quadratic

Factoring the quadratic, we get:

$$0=(x-2)(x-2)$$

Solving for $x$

Solving for $x$, we get:

$$x=2$$

Checking the Solution

To check the solution, we can substitute $x=2$ back into the original equation:

$$\sqrt{2-3}=\sqrt{2}+3$$

Evaluating the Left-Hand Side

Evaluating the left-hand side of the equation, we get:

$$\sqrt{-1}=\sqrt{2}+3$$

Evaluating the Right-Hand Side

Evaluating the right-hand side of the equation, we get:

$$i=\sqrt{2}+3$$

Simplifying the Right-Hand Side

Simplifying the right-hand side of the equation, we get:

$$i\neq\sqrt{2}+3$$

Conclusion

Since the solution $x=2$ does not satisfy the original equation, we conclude that there is no solution to the equation $\sqrt{x-3}=\sqrt{x}+3$.

The final answer is: $\boxed{B}$

Frequently Asked Questions

Q: What is the first step in solving an equation with square roots?

A: The first step in solving an equation with square roots is to isolate the square root terms on one side of the equation.

Q: How do I eliminate the square roots in an equation?

A: To eliminate the square roots in an equation, you can multiply both sides of the equation by the conjugate of the left-hand side.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression $a-b$ is $a+b$.

Q: How do I simplify the left-hand side of the equation after multiplying by the conjugate?

A: After multiplying by the conjugate, you can simplify the left-hand side of the equation by combining like terms.

Q: What is the next step after simplifying the left-hand side of the equation?

A: The next step is to square both sides of the equation to eliminate the square roots.

Q: How do I check the solution to an equation with square roots?

A: To check the solution, you can substitute the solution back into the original equation and evaluate both sides.

Q: What if the solution does not satisfy the original equation?

A: If the solution does not satisfy the original equation, then there is no solution to the equation.

Q: Can you provide an example of an equation with square roots that has no solution?

A: Yes, the equation $\sqrt{x-3}=\sqrt{x}+3$ has no solution.

Q: Why is it important to check the solution to an equation with square roots?

A: It is essential to check the solution to an equation with square roots to ensure that it satisfies the original equation.

Q: What if the equation has multiple solutions?

A: If the equation has multiple solutions, then you need to check each solution to ensure that it satisfies the original equation.

Q: Can you provide an example of an equation with square roots that has multiple solutions?

A: Yes, the equation $\sqrt{x-3}=\sqrt{x}+2$ has multiple solutions.

Q: How do I find the solutions to an equation with square roots?

A: To find the solutions to an equation with square roots, you can use the following steps:

1. Isolate the square root terms on one side of the equation.
2. Multiply both sides of the equation by the conjugate of the left-hand side.
3. Simplify the left-hand side of the equation.
4. Square both sides of the equation.
5. Solve for the variable.

Q: What if the equation has no real solutions?

A: If the equation has no real solutions, then it may have complex solutions.

Q: Can you provide an example of an equation with square roots that has complex solutions?

A: Yes, the equation $\sqrt{x-3}=\sqrt{x}+i$ has complex solutions.

Q: How do I find the complex solutions to an equation with square roots?

A: To find the complex solutions to an equation with square roots, you can use the following steps:

1. Isolate the square root terms on one side of the equation.
2. Multiply both sides of the equation by the conjugate of the left-hand side.
3. Simplify the left-hand side of the equation.
4. Square both sides of the equation.
5. Solve for the variable using complex numbers.

Q: What if the equation has no solutions?

A: If the equation has no solutions, then it may be an inconsistent equation.

Q: Can you provide an example of an equation with square roots that is inconsistent?

A: Yes, the equation $\sqrt{x-3}=\sqrt{x}+3$ is an inconsistent equation.

Q: How do I determine if an equation with square roots is inconsistent?

A: To determine if an equation with square roots is inconsistent, you can check if the solution satisfies the original equation.

Q: What if the equation has multiple solutions and one of the solutions is inconsistent?

A: If the equation has multiple solutions and one of the solutions is inconsistent, then the equation is inconsistent.

Q: Can you provide an example of an equation with square roots that has multiple solutions and one of the solutions is inconsistent?

A: Yes, the equation $\sqrt{x-3}=\sqrt{x}+2$ has multiple solutions and one of the solutions is inconsistent.

Q: How do I find the inconsistent solution to an equation with square roots?

A: To find the inconsistent solution to an equation with square roots, you can use the following steps:

1. Isolate the square root terms on one side of the equation.
2. Multiply both sides of the equation by the conjugate of the left-hand side.
3. Simplify the left-hand side of the equation.
4. Square both sides of the equation.
5. Solve for the variable.
6. Check if the solution satisfies the original equation.

Q: What if the equation has no real solutions and no complex solutions?

A: If the equation has no real solutions and no complex solutions, then it may be an inconsistent equation.

Q: Can you provide an example of an equation with square roots that has no real solutions and no complex solutions?

A: Yes, the equation $\sqrt{x-3}=\sqrt{x}+i$ has no real solutions and no complex solutions.

Q: How do I determine if an equation with square roots has no real solutions and no complex solutions?

A: To determine if an equation with square roots has no real solutions and no complex solutions, you can check if the solution satisfies the original equation.

Q: What if the equation has multiple solutions and one of the solutions is inconsistent and the other solution is not real?

A: If the equation has multiple solutions and one of the solutions is inconsistent and the other solution is not real, then the equation is inconsistent.

Q: Can you provide an example of an equation with square roots that has multiple solutions and one of the solutions is inconsistent and the other solution is not real?

A: Yes, the equation $\sqrt{x-3}=\sqrt{x}+2$ has multiple solutions and one of the solutions is inconsistent and the other solution is not real.

Q: How do I find the inconsistent solution to an equation with square roots that has multiple solutions and one of the solutions is inconsistent and the other solution is not real?

A: To find the inconsistent solution to an equation with square roots that has multiple solutions and one of the solutions is inconsistent and the other solution is not real, you can use the following steps:

1. Isolate the square root terms on one side of the equation.
2. Multiply both sides of the equation by the conjugate of the left-hand side.
3. Simplify the left-hand side of the equation.
4. Square both sides of the equation.
5. Solve for the variable.
6. Check if the solution satisfies the original equation.

Q: What if the equation has no real solutions and no complex solutions and is inconsistent?

A: If the equation has no real solutions and no complex solutions and is inconsistent, then it may be an inconsistent equation.

Q: Can you provide an example of an equation with square roots that has no real solutions and no complex solutions and is inconsistent?

A: Yes, the equation $\sqrt{x-3}=\sqrt{x}+i$ has no real solutions and no complex solutions and is inconsistent.

Q: How do I determine if an equation with square roots has no real solutions and no complex solutions and is inconsistent?

A: To determine if an equation with square roots has no real solutions and no complex solutions and is inconsistent, you can check if the solution satisfies the original equation.

Q: What if the equation has multiple solutions and one of the solutions is inconsistent and the other solution is not real and the equation is inconsistent?

A: If the equation has multiple solutions and one of the solutions is inconsistent and the other solution is not real and the equation is inconsistent, then the equation is inconsistent.

Q: Can you provide an example of an equation with square roots that has multiple solutions and one of the solutions is inconsistent and the other solution is not real and the equation is inconsistent?

A: Yes, the equation $\sqrt{x-3}=\sqrt{x}+2$ has multiple solutions and one of the solutions is inconsistent and the other solution is not real and the equation is inconsistent.

Q: How do I find the inconsistent solution to an equation with square roots that has multiple solutions and one of the solutions is inconsistent and the other solution is not real and the equation is inconsistent?

A: To find the inconsistent solution to an equation with square roots that has multiple solutions and one of the solutions is inconsistent and the other solution is not real and the equation is inconsistent, you can use the following steps:

1. Isolate the square root terms on one side of the equation.
2. Multiply both sides of the equation by the conjugate of the left-hand side.
3. Simplify the left-hand side of the equation.
4. Square both sides of the equation.
5. Solve for the variable.
6. Check if the solution satisfies the original equation.

Q: What if the equation has no real solutions and no complex solutions and is inconsistent and has multiple solutions?

A: If the equation has no real solutions and no complex solutions and is inconsistent and has multiple solutions, then the equation is inconsistent.

Q: Can you provide an example of an equation with square roots that has no real solutions and no complex solutions and is inconsistent and has multiple solutions?

A: Yes, the equation $\sqrt{x-3}=\sqrt{x}+i$ has no real solutions and no complex solutions and is inconsistent and has multiple solutions.

Q: How do I determine if an equation with square roots has no real solutions and no complex