Solve For All Possible Values Of \[$ X \$\].$\[ \sqrt{61 - 10x} = X - 7 \\]

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Introduction

Solving equations involving square roots can be a challenging task, especially when the equation involves a variable within the square root. In this article, we will explore how to solve the equation $\sqrt{61 - 10x} = x - 7$ for all possible values of $x$. This equation involves a square root, and our goal is to isolate the variable $x$ and find its possible values.

Step 1: Square both sides of the equation

To eliminate the square root, we can square both sides of the equation. This will allow us to simplify the equation and make it easier to solve. Squaring both sides of the equation gives us:

$$\left(\sqrt{61 - 10x}\right)^2 = (x - 7)^2$$

Step 2: Expand and simplify the equation

Expanding and simplifying the equation, we get:

$$61 - 10x = x^2 - 14x + 49$$

Step 3: Rearrange the equation

Rearranging the equation to get all the terms on one side, we get:

$$x^2 - 4x - 12 = 0$$

Step 4: Factor the quadratic equation

The quadratic equation $x^2 - 4x - 12 = 0$ can be factored as:

$$(x - 6)(x + 2) = 0$$

Step 5: Solve for $x$

To solve for $x$, we can set each factor equal to zero and solve for $x$. This gives us:

$$x - 6 = 0 \quad \text{or} \quad x + 2 = 0$$

Solving for $x$, we get:

$$x = 6 \quad \text{or} \quad x = -2$$

Step 6: Check the solutions

Before we can accept the solutions, we need to check if they are valid. We can do this by plugging the solutions back into the original equation and checking if it is true.

For $x = 6$, we get:

$$\sqrt{61 - 10(6)} = 6 - 7$$

$$\sqrt{61 - 60} = -1$$

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

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

For $x = -2$, we get:

$$\sqrt{61 - 10(-2)} = -2 - 7$$

$$\sqrt{61 + 20} = -9$$

$$\sqrt{81} = -9$$

This is not true, so $x = -2$ is not a valid solution.

Conclusion

Unfortunately, we were unable to find any valid solutions to the equation $\sqrt{61 - 10x} = x - 7$. This means that there are no values of $x$ that satisfy the equation.

Final Answer

The final answer is $\boxed{\text{No valid solutions}}$.

Discussion

Solving equations involving square roots can be a challenging task, especially when the equation involves a variable within the square root. In this article, we explored how to solve the equation $\sqrt{61 - 10x} = x - 7$ for all possible values of $x$. We squared both sides of the equation, expanded and simplified, factored the quadratic equation, and solved for $x$. Unfortunately, we were unable to find any valid solutions to the equation.

Related Topics

• Solving quadratic equations
• Square root equations
• Algebraic manipulations

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  

Introduction

Solving equations involving square roots can be a challenging task, especially when the equation involves a variable within the square root. In our previous article, we explored how to solve the equation $\sqrt{61 - 10x} = x - 7$ for all possible values of $x$. Unfortunately, we were unable to find any valid solutions to the equation. In this article, we will answer some frequently asked questions about solving equations involving square roots.

Q: What is the first step in solving an equation involving a square root?

A: The first step in solving an equation involving a square root is to square both sides of the equation. This will allow us to eliminate the square root and simplify the equation.

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. Squaring both sides of the equation allows us to get rid of the square root and make the equation easier to solve.

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 squaring both sides of the equation
• Not checking the solutions
• Not considering the domain of the square root function

Q: How do we check the solutions?

A: To check the solutions, we need to plug the solutions back into the original equation and check if it is true. If the solution is not true, then it is not a valid solution.

Q: What is the domain of the square root function?

A: The domain of the square root function is all real numbers greater than or equal to zero. This means that the expression inside the square root must be non-negative.

Q: Can we always find a solution to an equation involving a square root?

A: No, we cannot always find a solution to an equation involving a square root. Sometimes, the equation may have no valid solutions.

Q: How do we know if an equation involving a square root has no valid solutions?

A: We can determine if an equation involving a square root has no valid solutions by checking the solutions. If the solutions are not true, then the equation has no valid solutions.

Q: What are some real-world applications of solving equations involving square roots?

A: Solving equations involving square roots has many real-world applications, including:

• Physics: Solving equations involving square roots is used to calculate distances, velocities, and accelerations.
• Engineering: Solving equations involving square roots is used to design and optimize systems.
• Computer Science: Solving equations involving square roots is used in algorithms and data structures.

Conclusion

Solving equations involving square roots can be a challenging task, but with the right techniques and strategies, we can solve them. In this article, we answered some frequently asked questions about solving equations involving square roots. We hope that this article has been helpful in understanding how to solve equations involving square roots.

Final Answer

The final answer is $\boxed{\text{No valid solutions}}$.

Discussion

Solving equations involving square roots is an important topic in mathematics. It has many real-world applications and is used in various fields, including physics, engineering, and computer science. In this article, we explored some frequently asked questions about solving equations involving square roots.

Related Topics

• Solving quadratic equations
• Square root equations
• Algebraic manipulations

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton