Solve The Equation: X + 76 = X + 4 \sqrt{x+76} = X+4 X + 76 ​ = X + 4 Answer: X = □ X = \square X = □

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Introduction

Solving equations involving square roots can be a challenging task, especially when the equation is not straightforward. In this article, we will focus on solving the equation $\sqrt{x+76} = x+4$ and provide a step-by-step guide on how to arrive at the solution.

Understanding the Equation

The given equation is $\sqrt{x+76} = x+4$. This equation involves a square root, which means that the expression inside the square root must be non-negative. In other words, $x+76 \geq 0$, which implies that $x \geq -76$.

Isolating the Square Root

To solve the equation, we can start by isolating the square root on one side of the equation. We can do this by squaring both sides of the equation, which will eliminate the square root.

$\left(\sqrt{x+76}\right)^2 = (x+4)^2$

Expanding the Squared Expressions

When we expand the squared expressions, we get:

$x+76 = x^2 + 8x + 16$

Rearranging the Terms

To make it easier to solve the equation, we can rearrange the terms by subtracting $x+76$ from both sides of the equation.

$0 = x^2 + 7x - 60$

Factoring the Quadratic Expression

The quadratic expression $x^2 + 7x - 60$ can be factored as:

$(x+15)(x-4) = 0$

Solving for x

To find the values of $x$, we can set each factor equal to zero and solve for $x$.

$x+15 = 0 \Rightarrow x = -15$

$x-4 = 0 \Rightarrow x = 4$

Checking the Solutions

Before we can accept the solutions, we need to check if they satisfy the original equation. We can do this by plugging the solutions back into the original equation.

For $x = -15$:

$\sqrt{-15+76} = -15+4$

$\sqrt{61} = -11$

This is not true, so $x = -15$ is not a valid solution.

For $x = 4$:

$\sqrt{4+76} = 4+4$

$\sqrt{80} = 8$

This is true, so $x = 4$ is a valid solution.

Conclusion

In this article, we solved the equation $\sqrt{x+76} = x+4$ and found that the solution is $x = 4$. We also discussed the importance of checking the solutions to ensure that they satisfy the original equation.

Final Answer

The final answer is: $\boxed{4}$

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Introduction

In our previous article, we solved the equation $\sqrt{x+76} = x+4$ and found that the solution is $x = 4$. In this article, we will provide a Q&A section to address any questions or concerns that readers may have.

Q: What is the first step in solving the equation $\sqrt{x+76} = x+4$?

A: The first step in solving the equation is to isolate the square root on one side of the equation. We can do this by squaring both sides of the equation, which will eliminate the square root.

Q: Why do we need to check the solutions?

A: We need to check the solutions to ensure that they satisfy the original equation. If a solution does not satisfy the original equation, it is not a valid solution.

Q: What is the importance of checking the solutions?

A: Checking the solutions is crucial in ensuring that we have found the correct solution. If we do not check the solutions, we may end up with an incorrect solution.

Q: Can we use other methods to solve the equation $\sqrt{x+76} = x+4$?

A: Yes, we can use other methods to solve the equation. However, the method we used in our previous article is a common and efficient method for solving equations involving square roots.

Q: What is the difference between a valid solution and an invalid solution?

A: A valid solution is a solution that satisfies the original equation, while an invalid solution is a solution that does not satisfy the original equation.

Q: How do we determine if a solution is valid or invalid?

A: We can determine if a solution is valid or invalid by plugging the solution back into the original equation. If the solution satisfies the original equation, it is a valid solution. If the solution does not satisfy the original equation, it is an invalid solution.

Q: Can we use the same method to solve other equations involving square roots?

A: Yes, we can use the same method to solve other equations involving square roots. However, we need to be careful and ensure that we are using the correct method for the specific equation.

Q: What are some common mistakes to avoid when solving equations involving square roots?

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

• Not checking the solutions
• Not isolating the square root
• Not squaring both sides of the equation
• Not plugging the solutions back into the original equation

Conclusion

In this article, we provided a Q&A section to address any questions or concerns that readers may have. We discussed the importance of checking the solutions and provided tips on how to avoid common mistakes when solving equations involving square roots.

Final Answer

The final answer is: $\boxed{4}$