Solve For All Possible Values Of { X $} . . . { \sqrt{5x + 39} = X + 9 \}

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Introduction

In this article, we will delve into solving for all possible values of x in the given equation $\sqrt{5x + 39} = x + 9$. This equation involves a square root, making it a bit more complex than a standard linear equation. We will use algebraic techniques to isolate the variable x and find its possible values.

Understanding the Equation

The given equation is $\sqrt{5x + 39} = x + 9$. This equation states that the square root of $5x + 39$ is equal to $x + 9$. To solve for x, we need to get rid of the square root and isolate the variable x.

Step 1: Squaring Both Sides

One way to get rid of the square root is to square both sides of the equation. This will eliminate the square root, but it may introduce extraneous solutions. We will need to check our solutions later to ensure they are valid.

$\left(\sqrt{5x + 39}\right)^2 = (x + 9)^2$

Step 2: Expanding the Squared Terms

Now, we will expand the squared terms on both sides of the equation.

$5x + 39 = x^2 + 18x + 81$

Step 3: Rearranging the Terms

Next, we will rearrange the terms to form a quadratic equation.

$x^2 + 18x + 81 - 5x - 39 = 0$

Step 4: Simplifying the Equation

Now, we will simplify the equation by combining like terms.

$x^2 + 13x + 42 = 0$

Step 5: Factoring the Quadratic Equation

We can factor the quadratic equation to find its roots.

$(x + 6)(x + 7) = 0$

Step 6: Finding the Roots

Now, we will find the roots of the equation by setting each factor equal to zero.

$x + 6 = 0$ or $x + 7 = 0$

Step 7: Solving for x

Solving for x, we get:

$x = -6$ or $x = -7$

Step 8: Checking the Solutions

We need to check our solutions to ensure they are valid. We will substitute each solution back into the original equation to check if it is true.

For $x = -6$:

$\sqrt{5(-6) + 39} = -6 + 9$

$\sqrt{-15 + 39} = 3$

$\sqrt{24} = 3$

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

For $x = -7$:

$\sqrt{5(-7) + 39} = -7 + 9$

$\sqrt{-35 + 39} = 2$

$\sqrt{4} = 2$

This is true, so $x = -7$ is a valid solution.

Conclusion

In this article, we solved for all possible values of x in the equation $\sqrt{5x + 39} = x + 9$. We used algebraic techniques to isolate the variable x and find its possible values. We found that the only valid solution is $x = -7$.

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Introduction

In our previous article, we solved for all possible values of x in the equation $\sqrt{5x + 39} = x + 9$. We used algebraic techniques to isolate the variable x and find its possible values. In this article, we will answer some frequently asked questions related to the solution of the equation.

Q: What is the main concept behind solving the equation $\sqrt{5x + 39} = x + 9$?

A: The main concept behind solving the equation $\sqrt{5x + 39} = x + 9$ is to isolate the variable x and find its possible values. We used algebraic techniques, such as squaring both sides of the equation and factoring the quadratic equation, to solve for x.

Q: Why do we need to check the solutions?

A: We need to check the solutions to ensure they are valid. When we squared both sides of the equation, we may have introduced extraneous solutions. By substituting each solution back into the original equation, we can verify if it is true or not.

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

A: A valid solution is a solution that satisfies the original equation, while an extraneous solution is a solution that does not satisfy the original equation. In our case, $x = -6$ is an extraneous solution, while $x = -7$ is a valid solution.

Q: Can we use other methods to solve the equation $\sqrt{5x + 39} = x + 9$?

A: Yes, we can use other methods to solve the equation $\sqrt{5x + 39} = x + 9$. For example, we can use the method of substitution or the method of elimination. However, the method we used in our previous article is a common and efficient way to solve the equation.

Q: How do we know if the equation $\sqrt{5x + 39} = x + 9$ has any real solutions?

A: To determine if the equation $\sqrt{5x + 39} = x + 9$ has any real solutions, we need to check if the expression inside the square root is non-negative. In this case, $5x + 39 \geq 0$, which means $x \geq -7.8$. Therefore, the equation has real solutions for $x \geq -7.8$.

Q: Can we use the equation $\sqrt{5x + 39} = x + 9$ to model real-world problems?

A: Yes, we can use the equation $\sqrt{5x + 39} = x + 9$ to model real-world problems. For example, we can use this equation to model the growth of a population or the spread of a disease. However, we need to ensure that the equation is valid and makes sense in the context of the problem.

Q: How do we extend the solution to the equation $\sqrt{5x + 39} = x + 9$ to other similar equations?

A: To extend the solution to the equation $\sqrt{5x + 39} = x + 9$ to other similar equations, we need to identify the common pattern and use it to solve the new equation. For example, if we have the equation $\sqrt{ax + b} = x + c$, we can use the same method to solve for x.

Conclusion

In this article, we answered some frequently asked questions related to the solution of the equation $\sqrt{5x + 39} = x + 9$. We discussed the main concept behind solving the equation, the importance of checking the solutions, and how to extend the solution to other similar equations. We hope this article has provided valuable insights and helped you understand the solution to the equation.