Select The Correct Answer.What Is The Solution To The Equation − 2 X − 5 − 4 = X \sqrt{-2x - 5} - 4 = X − 2 X − 5 ​ − 4 = X ?A. -7 And -3 B. 3 And 7 C. -3 D. 7

Introduction

Equations involving square roots can be challenging to solve, but with the right approach, we can find the solutions. In this article, we will focus on solving the equation $\sqrt{-2x - 5} - 4 = x$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Step 1: Isolate the Square Root

The first step is to isolate the square root term on one side of the equation. We can do this by adding 4 to both sides of the equation:

$$\sqrt{-2x - 5} = x + 4$$

Step 2: Square Both Sides

To eliminate the square root, we can square both sides of the equation:

$$(-2x - 5) = (x + 4)^2$$

Step 3: Expand the Right Side

We can expand the right side of the equation using the formula $(a + b)^2 = a^2 + 2ab + b^2$:

$$(-2x - 5) = x^2 + 8x + 16$$

Step 4: Move All Terms to One Side

We can move all terms to one side of the equation by subtracting $x^2 + 8x + 16$ from both sides:

$$-2x - 5 - x^2 - 8x - 16 = 0$$

Step 5: Simplify the Equation

We can simplify the equation by combining like terms:

$$-x^2 - 10x - 21 = 0$$

Step 6: Factor the Quadratic Equation

We can factor the quadratic equation:

$$-(x^2 + 10x + 21) = 0$$

$$(x + 7)(x + 3) = 0$$

Step 7: Solve for x

We can solve for x by setting each factor equal to zero:

$$x + 7 = 0 \Rightarrow x = -7$$

$$x + 3 = 0 \Rightarrow x = -3$$

Conclusion

We have successfully solved the equation $\sqrt{-2x - 5} - 4 = x$. The solutions are $x = -7$ and $x = -3$. We can verify these solutions by plugging them back into the original equation.

Verification

We can verify the solutions by plugging them back into the original equation:

$$\sqrt{-2(-7) - 5} - 4 = -7$$

$$\sqrt{14 - 5} - 4 = -7$$

$$\sqrt{9} - 4 = -7$$

$$3 - 4 = -7$$

$$-1 = -7$$

This is not true, so $x = -7$ is not a solution.

$$\sqrt{-2(-3) - 5} - 4 = -3$$

$$\sqrt{6 - 5} - 4 = -3$$

$$\sqrt{1} - 4 = -3$$

$$1 - 4 = -3$$

$$-3 = -3$$

This is true, so $x = -3$ is a solution.

Final Answer

The final answer is $\boxed{-3}$.

Note

Q: What is the main concept behind solving the equation $\sqrt{-2x - 5} - 4 = x$?

A: The main concept behind solving this equation is to isolate the square root term, square both sides, and then solve for x using algebraic manipulations.

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

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

Q: What is the purpose of squaring both sides of the equation?

A: The purpose of squaring both sides of the equation is to eliminate the square root term. When we square both sides, we get rid of the square root and are left with a polynomial equation that we can solve.

Q: How do we expand the right side of the equation?

A: We expand the right side of the equation using the formula $(a + b)^2 = a^2 + 2ab + b^2$. This formula allows us to expand the squared term and simplify the equation.

Q: What is the significance of factoring the quadratic equation?

A: Factoring the quadratic equation allows us to find the solutions to the equation. By factoring the equation, we can set each factor equal to zero and solve for x.

Q: Why do we need to verify the solutions?

A: We need to verify the solutions because we may have made an error in our calculations. By plugging the solutions back into the original equation, we can check if they are true or not.

Q: What is the final answer to the equation $\sqrt{-2x - 5} - 4 = x$?

A: The final answer to the equation $\sqrt{-2x - 5} - 4 = x$ is $x = -3$.

Q: What are the common mistakes to avoid when solving equations with square roots?

A: Some common mistakes to avoid when solving equations with square roots include:

• Not isolating the square root term
• Not squaring both sides of the equation
• Not expanding the right side of the equation correctly
• Not factoring the quadratic equation correctly
• Not verifying the solutions

Q: How can we apply the concepts learned in this article to other equations?

A: We can apply the concepts learned in this article to other equations by following the same steps:

• Isolate the square root term
• Square both sides of the equation
• Expand the right side of the equation
• Factor the quadratic equation
• Verify the solutions

By following these steps, we can solve equations with square roots and other types of equations.

Conclusion

In this article, we have discussed how to solve the equation $\sqrt{-2x - 5} - 4 = x$. We have also answered some frequently asked questions about solving equations with square roots. By following the steps outlined in this article, we can solve equations with square roots and other types of equations.