Solve The Equation:$ \sqrt{x+10} - 4 = -x - 2 $

Introduction

In this article, we will delve into the world of algebra and solve a complex equation step by step. The equation we will be solving is: $\sqrt{x+10} - 4 = -x - 2$. This equation involves square roots and linear terms, making it a challenging problem to solve. However, with a clear understanding of the steps involved, we can break down the solution into manageable parts.

Understanding the Equation

The given equation is: $\sqrt{x+10} - 4 = -x - 2$. To solve this equation, we need to isolate the variable $x$. The first step is to simplify the equation by getting rid of the square root. We can do this by adding $4$ to both sides of the equation.

Simplifying the Equation

$\sqrt{x+10} - 4 + 4 = -x - 2 + 4$

This simplifies to:

$\sqrt{x+10} = -x + 2$

Isolating the Square Root

To isolate the square root, we can square both sides of the equation. This will eliminate the square root and allow us to solve for $x$.

Squaring Both Sides

$(\sqrt{x+10})^2 = (-x + 2)^2$

This simplifies to:

$x + 10 = x^2 - 4x + 4$

Expanding and Simplifying

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

$x + 10 = x^2 - 4x + 4$

Subtracting $x$ from both sides, we get:

$10 = x^2 - 5x + 4$

Rearranging the Equation

Rearranging the equation to put it in standard quadratic form, we get:

$x^2 - 5x - 6 = 0$

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$, $b = -5$, and $c = -6$. Plugging these values into the formula, we get:

$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-6)}}{2(1)}$

Simplifying the expression, we get:

$x = \frac{5 \pm \sqrt{25 + 24}}{2}$

$x = \frac{5 \pm \sqrt{49}}{2}$

$x = \frac{5 \pm 7}{2}$

Finding the Solutions

Simplifying the expression, we get two possible solutions:

$x = \frac{5 + 7}{2}$

$x = \frac{5 - 7}{2}$

$x = 6$

$x = -1$

Checking the Solutions

To check the solutions, we can plug them back into the original equation. If the equation holds true, then the solution is valid.

Checking $x = 6$

$\sqrt{6+10} - 4 = -6 - 2$

$\sqrt{16} - 4 = -8$

$4 - 4 = -8$

$0 = -8$

This is not true, so $x = 6$ is not a valid solution.

Checking $x = -1$

$\sqrt{-1+10} - 4 = -(-1) - 2$

$\sqrt{9} - 4 = 1 - 2$

$3 - 4 = -1$

This is true, so $x = -1$ is a valid solution.

Conclusion

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Introduction

In our previous article, we solved the equation $\sqrt{x+10} - 4 = -x - 2$ step by step. However, we understand that some readers may still have questions about the solution. In this article, we will address some of the most frequently asked questions about solving the equation.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to simplify the equation by getting rid of the square root. We can do this by adding $4$ to both sides of the equation.

Q: Why do we need to isolate the square root?

A: We need to isolate the square root because it is the only term that contains the variable $x$. By isolating the square root, we can eliminate it and solve for $x$.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do we use the quadratic formula to solve the equation?

A: To use the quadratic formula to solve the equation, we need to identify the values of $a$, $b$, and $c$. In this case, $a = 1$, $b = -5$, and $c = -6$. We then plug these values into the formula and simplify the expression.

Q: What are the two possible solutions to the equation?

A: The two possible solutions to the equation are $x = 6$ and $x = -1$.

Q: How do we check the solutions?

A: To check the solutions, we plug them back into the original equation. If the equation holds true, then the solution is valid.

Q: Why is $x = 6$ not a valid solution?

A: $x = 6$ is not a valid solution because when we plug it back into the original equation, we get a false statement.

Q: Why is $x = -1$ a valid solution?

A: $x = -1$ is a valid solution because when we plug it back into the original equation, we get a true statement.

Q: What is the final solution to the equation?

A: The final solution to the equation is $x = -1$.

Conclusion

In this article, we addressed some of the most frequently asked questions about solving the equation $\sqrt{x+10} - 4 = -x - 2$. We provided step-by-step explanations and examples to help readers understand the solution. We hope that this article has been helpful in clarifying any doubts that readers may have had.

Additional Resources

For more information on solving quadratic equations, please refer to the following resources:

• Quadratic Formula
• Solving Quadratic Equations
• Quadratic Equations

Final Thoughts

Solving the equation $\sqrt{x+10} - 4 = -x - 2$ requires patience, persistence, and a clear understanding of the steps involved. We hope that this article has been helpful in providing a clear and concise explanation of the solution. If you have any further questions or concerns, please do not hesitate to contact us.