Select The Correct Answer From Each Drop-down Menu.Consider This Equation: \[$\sqrt{x-1} - 5 = X - 8\$\]The Equation Has \[$\square\$\] And \[$\square\$\].A Valid Solution For \[$x\$\] Is \[$\square\$\].

Introduction

In this article, we will be solving a quadratic equation that involves a square root. The equation is given as: $\sqrt{x-1} - 5 = x - 8$. Our goal is to find the value of $x$ that satisfies this equation. We will break down the solution into smaller steps and provide a clear explanation of each step.

Step 1: Isolate the Square Root

The first step in solving this equation is to isolate the square root term. We can do this by adding 5 to both sides of the equation. This gives us:

$\sqrt{x-1} = x - 8 + 5$

Simplifying the right-hand side, we get:

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

Step 2: Square Both Sides

The next step is to square both sides of the equation. This will eliminate the square root term. We get:

$(\sqrt{x-1})^2 = (x - 3)^2$

Simplifying both sides, we get:

$x - 1 = x^2 - 6x + 9$

Step 3: Rearrange the Equation

Now, we need to rearrange the equation to get all the terms on one side. We can do this by subtracting $x$ from both sides and adding 1 to both sides. This gives us:

$0 = x^2 - 7x + 10$

Step 4: Factor the Quadratic

The next step is to factor the quadratic equation. We can do this by finding two numbers whose product is 10 and whose sum is -7. These numbers are -5 and -2. Therefore, we can factor the quadratic as:

$0 = (x - 5)(x - 2)$

Step 5: Solve for x

Now, we can solve for $x$ by setting each factor equal to zero. This gives us:

$x - 5 = 0$ or $x - 2 = 0$

Solving for $x$, we get:

$x = 5$ or $x = 2$

Conclusion

In this article, we solved a quadratic equation that involved a square root. We broke down the solution into smaller steps and provided a clear explanation of each step. We found that the equation has two solutions: $x = 5$ and $x = 2$. A valid solution for $x$ is $\boxed{5}$ and $\boxed{2}$.

Discussion

This equation is a quadratic equation that involves a square root. The square root term is isolated in the first step, and then squared in the second step. The resulting quadratic equation is then factored in the fourth step. The solutions to the equation are found in the fifth step.

Key Takeaways

• To solve a quadratic equation that involves a square root, we need to isolate the square root term and then square both sides.
• The resulting quadratic equation can be factored to find the solutions.
• The solutions to the equation are found by setting each factor equal to zero.

Additional Resources

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

• Khan Academy: Solving Quadratic Equations
• Mathway: Solving Quadratic Equations
• Wolfram Alpha: Solving Quadratic Equations

Final Answer

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Introduction

In our previous article, we solved a quadratic equation that involved a square root. We broke down the solution into smaller steps and provided a clear explanation of each step. In this article, we will answer some frequently asked questions about solving quadratic equations that involve a square root.

Q: What is the first step in solving a quadratic equation that involves a square root?

A: The first step in solving a quadratic equation that involves a square root is to isolate the square root term. This can be done by adding or subtracting a constant from both sides of the equation.

Q: How do I isolate the square root term?

A: To isolate the square root term, you need to get it by itself on one side of the equation. This can be done by adding or subtracting a constant from both sides of the equation. For example, if the equation is $\sqrt{x-1} - 5 = x - 8$, you can add 5 to both sides to get $\sqrt{x-1} = x - 3$.

Q: What is the next step after isolating the square root term?

A: The next step after isolating the square root term is to square both sides of the equation. This will eliminate the square root term and give you a quadratic equation.

Q: How do I square both sides of the equation?

A: To square both sides of the equation, you need to multiply both sides by themselves. For example, if the equation is $\sqrt{x-1} = x - 3$, you can square both sides to get $(\sqrt{x-1})^2 = (x - 3)^2$.

Q: What is the next step after squaring both sides of the equation?

A: The next step after squaring both sides of the equation is to simplify the equation and rearrange it to get all the terms on one side. This will give you a quadratic equation that you can factor.

Q: How do I simplify the equation and rearrange it?

A: To simplify the equation and rearrange it, you need to combine like terms and get all the terms on one side of the equation. For example, if the equation is $x - 1 = x^2 - 6x + 9$, you can simplify it by combining like terms to get $0 = x^2 - 7x + 10$.

Q: What is the next step after simplifying the equation and rearranging it?

A: The next step after simplifying the equation and rearranging it is to factor the quadratic equation. This will give you two binomial factors that you can set equal to zero to find the solutions.

Q: How do I factor the quadratic equation?

A: To factor the quadratic equation, you need to find two binomial factors that multiply to give the original quadratic expression. For example, if the equation is $0 = x^2 - 7x + 10$, you can factor it as $(x - 5)(x - 2)$.

Q: What is the final step in solving a quadratic equation that involves a square root?

A: The final step in solving a quadratic equation that involves a square root is to set each factor equal to zero and solve for $x$. This will give you the solutions to the equation.

Q: How do I set each factor equal to zero and solve for $x$?

A: To set each factor equal to zero and solve for $x$, you need to set each binomial factor equal to zero and solve for $x$. For example, if the equation is $(x - 5)(x - 2) = 0$, you can set each factor equal to zero to get $x - 5 = 0$ or $x - 2 = 0$. Solving for $x$, you get $x = 5$ or $x = 2$.

Conclusion

In this article, we answered some frequently asked questions about solving quadratic equations that involve a square root. We covered the steps involved in solving such equations, from isolating the square root term to setting each factor equal to zero and solving for $x$. We hope that this article has been helpful in clarifying the process of solving quadratic equations that involve a square root.

Key Takeaways

• To solve a quadratic equation that involves a square root, you need to isolate the square root term and then square both sides.
• The resulting quadratic equation can be factored to find the solutions.
• The solutions to the equation are found by setting each factor equal to zero and solving for $x$.

Additional Resources

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

• Khan Academy: Solving Quadratic Equations
• Mathway: Solving Quadratic Equations
• Wolfram Alpha: Solving Quadratic Equations

Final Answer

The final answer is: $\boxed{5}$ and $\boxed{2}$.