Select All The Solutions To $\sqrt{(x-2)^2}=\sqrt{-16}$.A. $x=6$ B. $x=-2$ C. $x=-6$ D. $x=2+4i$ E. $x=2+2i$ F. $x=2-2i$ G. $x=2-4i$

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Introduction

Solving equations involving square roots can be a challenging task, especially when dealing with complex numbers. In this article, we will explore the solutions to the equation $\sqrt{(x-2)^2}=\sqrt{-16}$ and provide a step-by-step guide on how to arrive at the correct solutions.

Understanding the Equation

The given equation is $\sqrt{(x-2)^2}=\sqrt{-16}$. To begin solving this equation, we need to understand the properties of square roots and how they interact with complex numbers.

The square root of a negative number is an imaginary number, which can be represented as $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, defined as the square root of $-1$. In this case, we have $\sqrt{-16}$, which can be simplified to $4i$.

Simplifying the Equation

Now that we have simplified the right-hand side of the equation, we can rewrite it as $\sqrt{(x-2)^2}=4i$. To eliminate the square root, we can square both sides of the equation, resulting in $(x-2)^2=16i^2$.

Simplifying the Left-Hand Side

The left-hand side of the equation is $(x-2)^2$, which can be expanded as $x^2-4x+4$. Substituting this into the equation, we get $x^2-4x+4=16i^2$.

Simplifying the Right-Hand Side

The right-hand side of the equation is $16i^2$, which can be simplified to $-16$. Substituting this into the equation, we get $x^2-4x+4=-16$.

Rearranging the Equation

To make it easier to solve the equation, we can rearrange it by adding $16$ to both sides, resulting in $x^2-4x+20=0$.

Solving the Quadratic Equation

The equation $x^2-4x+20=0$ is a quadratic equation, which can be solved using the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.

In this case, $a=1$, $b=-4$, and $c=20$. Substituting these values into the quadratic formula, we get $x=\frac{-(-4)\pm\sqrt{(-4)^2-4(1)(20)}}{2(1)}$.

Simplifying the Quadratic Formula

Simplifying the quadratic formula, we get $x=\frac{4\pm\sqrt{16-80}}{2}$. This can be further simplified to $x=\frac{4\pm\sqrt{-64}}{2}$.

Simplifying the Square Root

The square root of $-64$ can be simplified to $8i$. Substituting this into the equation, we get $x=\frac{4\pm8i}{2}$.

Simplifying the Final Answer

Simplifying the final answer, we get $x=2\pm4i$.

Conclusion

In conclusion, the solutions to the equation $\sqrt{(x-2)^2}=\sqrt{-16}$ are $x=2+4i$ and $x=2-4i$. These solutions can be verified by substituting them back into the original equation.

Final Answer

The final answer is: $\boxed{D, G}$

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Introduction

In our previous article, we explored the solutions to the equation $\sqrt{(x-2)^2}=\sqrt{-16}$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into the solutions.

Q: What is the first step in solving the equation $\sqrt{(x-2)^2}=\sqrt{-16}$?

A: The first step in solving the equation is to simplify the right-hand side of the equation, which is $\sqrt{-16}$. This can be simplified to $4i$.

Q: Why do we need to square both sides of the equation?

A: We need to square both sides of the equation to eliminate the square root. This is because the square root of a negative number is an imaginary number, and squaring both sides allows us to work with real numbers.

Q: How do we simplify the left-hand side of the equation?

A: The left-hand side of the equation is $(x-2)^2$, which can be expanded as $x^2-4x+4$. This is a quadratic expression, and we can substitute it into the equation.

Q: What is the final form of the equation after simplifying?

A: The final form of the equation is $x^2-4x+20=0$. This is a quadratic equation, and we can solve it using the quadratic formula.

Q: How do we solve the quadratic equation?

A: We can solve the quadratic equation using the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. In this case, $a=1$, $b=-4$, and $c=20$.

Q: What are the solutions to the quadratic equation?

A: The solutions to the quadratic equation are $x=2\pm4i$. These are the final solutions to the original equation.

Q: How do we verify the solutions?

A: We can verify the solutions by substituting them back into the original equation. If the solutions satisfy the equation, then they are the correct solutions.

Q: What are the final solutions to the equation $\sqrt{(x-2)^2}=\sqrt{-16}$?

A: The final solutions to the equation are $x=2+4i$ and $x=2-4i$.

Q: Why are these solutions correct?

A: These solutions are correct because they satisfy the original equation. When we substitute them back into the equation, we get a true statement.

Q: What is the significance of the solutions?

A: The solutions $x=2+4i$ and $x=2-4i$ are significant because they represent the points on the complex plane where the equation is satisfied. These points are the solutions to the equation.

Q: How do we interpret the solutions in the context of the original equation?

A: We can interpret the solutions as the values of $x$ that make the equation true. In other words, these values of $x$ satisfy the equation and make it true.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{D, G}$, which represents the solutions $x=2+4i$ and $x=2-4i$.

Conclusion

In conclusion, the Q&A section provides additional insights and clarifies any doubts about the solutions to the equation $\sqrt{(x-2)^2}=\sqrt{-16}$. The final solutions to the equation are $x=2+4i$ and $x=2-4i$, and these solutions can be verified by substituting them back into the original equation.