Select The Correct Answer.What Is A Solution To This Equation? { (1-3x)^{\frac{1}{2}} - 1 = X$}$A. -5 B. 0 C. -1 D. 1

Introduction

Equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific equation, $(1-3x)^{\frac{1}{2}} - 1 = x$, and provide a step-by-step guide to finding the correct solution.

Understanding the Equation

The given equation is a quadratic equation, which means it contains a squared variable. The equation is $(1-3x)^{\frac{1}{2}} - 1 = x$. To solve this equation, we need to isolate the variable $x$.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by getting rid of the square root. We can do this by isolating the square root term and then squaring both sides of the equation.

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

To simplify the equation, we can add 1 to both sides:

(1-3x)^{\frac{1}{2}} = x + 1

Step 2: Square Both Sides

Now that we have isolated the square root term, we can square both sides of the equation to get rid of the square root.

(1-3x)^{\frac{1}{2}} = x + 1

Squaring both sides gives us:

1-3x = (x + 1)^2

Step 3: Expand the Right Side

To simplify the equation further, we need to expand the right side of the equation.

1-3x = (x + 1)^2

Expanding the right side gives us:

1-3x = x^2 + 2x + 1

Step 4: Simplify the Equation

Now that we have expanded the right side of the equation, we can simplify it by combining like terms.

1-3x = x^2 + 2x + 1

Simplifying the equation gives us:

-3x = x^2 + 2x

Step 5: Rearrange the Equation

To solve for $x$, we need to rearrange the equation so that all the terms are on one side.

-3x = x^2 + 2x

Rearranging the equation gives us:

x^2 + 5x = 0

Step 6: Factor the Equation

Now that we have rearranged the equation, we can factor it to find the solutions.

x^2 + 5x = 0

Factoring the equation gives us:

x(x + 5) = 0

Step 7: Solve for $x$

To find the solutions, we need to set each factor equal to zero and solve for $x$.

x(x + 5) = 0

Setting each factor equal to zero gives us:

x = 0
x + 5 = 0

Solving for $x$ gives us:

x = 0
x = -5

Conclusion

In this article, we have solved the equation $(1-3x)^{\frac{1}{2}} - 1 = x$ using a step-by-step guide. We simplified the equation, squared both sides, expanded the right side, simplified the equation, rearranged the equation, factored the equation, and finally solved for $x$. The solutions to the equation are $x = 0$ and $x = -5$.

Answer

The correct answer is:

• A. -5

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Introduction

In our previous article, we solved the equation $(1-3x)^{\frac{1}{2}} - 1 = x$ using a step-by-step guide. In this article, we will answer some frequently asked questions related to solving the equation.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to simplify it by getting rid of the square root. We can do this by isolating the square root term and then squaring both sides of 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 get rid of the square root. This is because the square root of a number is equal to the number raised to the power of 1/2. By squaring both sides, we can eliminate the square root and simplify the equation.

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

A: To expand the right side of the equation, we need to multiply out the terms inside the parentheses. In this case, we have $(x + 1)^2$, which expands to $x^2 + 2x + 1$.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is an equation that contains a squared variable, while a linear equation is an equation that contains only a linear term. In this case, the equation $(1-3x)^{\frac{1}{2}} - 1 = x$ is a quadratic equation because it contains the squared term $(1-3x)^{\frac{1}{2}}$.

Q: How do we factor a quadratic equation?

A: To factor a quadratic equation, we need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term. In this case, we have the equation $x^2 + 5x = 0$, which factors to $x(x + 5) = 0$.

Q: What is the final solution to the equation?

A: The final solution to the equation is $x = 0$ and $x = -5$. These are the two values of $x$ that satisfy the equation $(1-3x)^{\frac{1}{2}} - 1 = x$.

Q: Why is it important to check our work when solving an equation?

A: It is essential to check our work when solving an equation to ensure that we have found the correct solution. This involves plugging the solution back into the original equation to verify that it is true.

Q: What are some common mistakes to avoid when solving an equation?

A: Some common mistakes to avoid when solving an equation include:

• Not simplifying the equation enough
• Not squaring both sides of the equation
• Not expanding the right side of the equation
• Not factoring the equation correctly
• Not checking our work

Conclusion

In this article, we have answered some frequently asked questions related to solving the equation $(1-3x)^{\frac{1}{2}} - 1 = x$. We have covered topics such as simplifying the equation, squaring both sides, expanding the right side, factoring the equation, and checking our work. By following these steps and avoiding common mistakes, we can ensure that we find the correct solution to the equation.

Additional Resources

For more information on solving equations, we recommend the following resources:

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

By following these resources and practicing solving equations, you can become proficient in solving equations and apply this skill to a wide range of mathematical problems.