Solve The Equation: ( X 2 − 1 ) ( 7 X + 1 ) = 0 \left(x^2-1\right)(7x+1)=0 ( X 2 − 1 ) ( 7 X + 1 ) = 0 Select The Correct Choice Below And, If Necessary, Fill In The Blank:A. The Solution Set Is {___}. (Use A Comma To Separate Answers As Needed.)B. The Solution Set Is { \varnothing$}$.

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Introduction

In this article, we will focus on solving a quadratic equation of the form (x2−1)(7x+1)=0\left(x^2-1\right)(7x+1)=0. This equation involves the product of two binomials, and we will use the zero-product property to find the solutions. The zero-product property states that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero.

Understanding the Zero-Product Property

The zero-product property is a fundamental concept in algebra that allows us to solve equations involving the product of two or more factors. If we have an equation of the form ab=0ab=0, where aa and bb are expressions, then we can conclude that either a=0a=0 or b=0b=0. This property is based on the fact that the product of two numbers is zero only if at least one of the numbers is zero.

Applying the Zero-Product Property to the Given Equation

Now, let's apply the zero-product property to the given equation (x2−1)(7x+1)=0\left(x^2-1\right)(7x+1)=0. Using the zero-product property, we can conclude that either (x2−1)=0\left(x^2-1\right)=0 or (7x+1)=0(7x+1)=0. This gives us two separate equations to solve:

  1. (x2−1)=0\left(x^2-1\right)=0
  2. (7x+1)=0(7x+1)=0

Solving the First Equation

To solve the first equation (x2−1)=0\left(x^2-1\right)=0, we can start by factoring the quadratic expression x2−1x^2-1. We can rewrite this expression as (x−1)(x+1)=0(x-1)(x+1)=0. Now, we can apply the zero-product property again to conclude that either (x−1)=0(x-1)=0 or (x+1)=0(x+1)=0. This gives us two possible solutions:

  1. (x−1)=0⇒x=1(x-1)=0 \Rightarrow x=1
  2. (x+1)=0⇒x=−1(x+1)=0 \Rightarrow x=-1

Solving the Second Equation

To solve the second equation (7x+1)=0(7x+1)=0, we can start by isolating the variable xx. We can subtract 1 from both sides of the equation to get 7x=−17x=-1. Then, we can divide both sides of the equation by 7 to get x=−17x=-\frac{1}{7}.

Combining the Solutions

Now that we have solved both equations, we can combine the solutions to find the complete solution set. The solution set is the set of all values of xx that satisfy the original equation. In this case, the solution set is the union of the two solution sets:

{1,−1,−17}\{1, -1, -\frac{1}{7}\}

Conclusion

In this article, we have solved the equation (x2−1)(7x+1)=0\left(x^2-1\right)(7x+1)=0 using the zero-product property. We have found that the solution set is {1,−1,−17}\{1, -1, -\frac{1}{7}\}. This solution set represents all the values of xx that satisfy the original equation.

Final Answer

The final answer is: {1,−1,−17}\boxed{\{1, -1, -\frac{1}{7}\}}

Introduction

In our previous article, we solved the equation (x2−1)(7x+1)=0\left(x^2-1\right)(7x+1)=0 using the zero-product property. We found that the solution set is {1,−1,−17}\{1, -1, -\frac{1}{7}\}. In this article, we will answer some frequently asked questions related to solving this equation.

Q&A

Q: What is the zero-product property?

A: The zero-product property is a fundamental concept in algebra that allows us to solve equations involving the product of two or more factors. If we have an equation of the form ab=0ab=0, where aa and bb are expressions, then we can conclude that either a=0a=0 or b=0b=0.

Q: How do we apply the zero-product property to the given equation?

A: To apply the zero-product property to the given equation (x2−1)(7x+1)=0\left(x^2-1\right)(7x+1)=0, we can conclude that either (x2−1)=0\left(x^2-1\right)=0 or (7x+1)=0(7x+1)=0. This gives us two separate equations to solve.

Q: What are the steps to solve the first equation?

A: To solve the first equation (x2−1)=0\left(x^2-1\right)=0, we can start by factoring the quadratic expression x2−1x^2-1. We can rewrite this expression as (x−1)(x+1)=0(x-1)(x+1)=0. Then, we can apply the zero-product property again to conclude that either (x−1)=0(x-1)=0 or (x+1)=0(x+1)=0. This gives us two possible solutions: x=1x=1 and x=−1x=-1.

Q: What are the steps to solve the second equation?

A: To solve the second equation (7x+1)=0(7x+1)=0, we can start by isolating the variable xx. We can subtract 1 from both sides of the equation to get 7x=−17x=-1. Then, we can divide both sides of the equation by 7 to get x=−17x=-\frac{1}{7}.

Q: How do we combine the solutions?

A: To combine the solutions, we can find the union of the two solution sets. The solution set is the set of all values of xx that satisfy the original equation. In this case, the solution set is {1,−1,−17}\{1, -1, -\frac{1}{7}\}.

Q: What is the final answer?

A: The final answer is {1,−1,−17}\boxed{\{1, -1, -\frac{1}{7}\}}.

Q: Can we use other methods to solve this equation?

A: Yes, we can use other methods to solve this equation, such as using the quadratic formula or factoring. However, the zero-product property is a more straightforward and efficient method to solve this equation.

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

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

  • Not applying the zero-product property correctly
  • Not factoring the quadratic expression correctly
  • Not isolating the variable correctly
  • Not combining the solutions correctly

Conclusion

In this article, we have answered some frequently asked questions related to solving the equation (x2−1)(7x+1)=0\left(x^2-1\right)(7x+1)=0. We have provided step-by-step solutions and explanations to help readers understand the concept of the zero-product property and how to apply it to solve this equation.

Final Answer

The final answer is: {1,−1,−17}\boxed{\{1, -1, -\frac{1}{7}\}}