Solve The Following Equation. Write Your Answer In Reduced Fraction Form, If Necessary. ( 9 X + 1 ) ( X + 1 ) = 0 (9x + 1)(x + 1) = 0 ( 9 X + 1 ) ( X + 1 ) = 0 $x =$ (Separate Multiple Entries With Commas.)

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Introduction

In mathematics, solving equations is a crucial skill that helps us find the value of unknown variables. In this article, we will focus on solving a quadratic equation of the form (9x+1)(x+1)=0(9x + 1)(x + 1) = 0. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Equation

The given equation is a quadratic equation, which is a polynomial equation of degree two. It is in the form of (ax+b)(cx+d)=0(ax + b)(cx + d) = 0, where aa, bb, cc, and dd are constants. In this case, we have (9x+1)(x+1)=0(9x + 1)(x + 1) = 0. Our goal is to find the value of xx that satisfies this equation.

Step 1: Apply the Zero Product Property

The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In this case, we can apply the zero product property to the given equation:

(9x+1)(x+1)=0(9x + 1)(x + 1) = 0

This means that either (9x+1)=0(9x + 1) = 0 or (x+1)=0(x + 1) = 0.

Step 2: Solve the First Factor

Let's start by solving the first factor, (9x+1)=0(9x + 1) = 0. To do this, we can subtract 1 from both sides of the equation:

9x=−19x = -1

Next, we can divide both sides of the equation by 9:

x=−19x = -\frac{1}{9}

Step 3: Solve the Second Factor

Now, let's solve the second factor, (x+1)=0(x + 1) = 0. To do this, we can subtract 1 from both sides of the equation:

x=−1x = -1

Conclusion

In this article, we have solved the quadratic equation (9x+1)(x+1)=0(9x + 1)(x + 1) = 0 using the zero product property. We have found that the solutions to the equation are x=−19x = -\frac{1}{9} and x=−1x = -1. These solutions are in reduced fraction form, as required.

Final Answer

The final answer to the equation is:

x=−19,−1x = -\frac{1}{9}, -1

Why is this Important?

Solving quadratic equations is an essential skill in mathematics, and it has many real-world applications. For example, in physics, quadratic equations are used to model the motion of objects under the influence of gravity. In engineering, quadratic equations are used to design and optimize systems, such as bridges and buildings.

Tips and Tricks

Here are some tips and tricks to help you solve quadratic equations:

  • Always start by applying the zero product property.
  • Use algebraic manipulations to simplify the equation.
  • Check your solutions by plugging them back into the original equation.
  • Use a calculator or computer software to check your solutions.

Common Mistakes

Here are some common mistakes to avoid when solving quadratic equations:

  • Failing to apply the zero product property.
  • Not simplifying the equation before solving.
  • Not checking the solutions by plugging them back into the original equation.
  • Not using a calculator or computer software to check the solutions.

Real-World Applications

Quadratic equations have many real-world applications, including:

  • Physics: Modeling the motion of objects under the influence of gravity.
  • Engineering: Designing and optimizing systems, such as bridges and buildings.
  • Computer Science: Solving problems in computer graphics and game development.
  • Economics: Modeling the behavior of economic systems.

Conclusion

Introduction

In our previous article, we solved the quadratic equation (9x+1)(x+1)=0(9x + 1)(x + 1) = 0 using the zero product property. In this article, we will answer some frequently asked questions about quadratic equations.

Q: What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two, which means it has a highest power of two. It is in the form of ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

Q: How do I solve a quadratic equation?

To solve a quadratic equation, you can use the following steps:

  1. Apply the zero product property.
  2. Simplify the equation.
  3. Solve for the variable.

Q: What is the zero product property?

The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero.

Q: How do I apply the zero product property?

To apply the zero product property, you can set each factor equal to zero and solve for the variable.

Q: What are some common mistakes to avoid when solving quadratic equations?

Some common mistakes to avoid when solving quadratic equations include:

  • Failing to apply the zero product property.
  • Not simplifying the equation before solving.
  • Not checking the solutions by plugging them back into the original equation.
  • Not using a calculator or computer software to check the solutions.

Q: How do I check my solutions?

To check your solutions, you can plug them back into the original equation and see if they are true.

Q: What are some real-world applications of quadratic equations?

Quadratic equations have many real-world applications, including:

  • Physics: Modeling the motion of objects under the influence of gravity.
  • Engineering: Designing and optimizing systems, such as bridges and buildings.
  • Computer Science: Solving problems in computer graphics and game development.
  • Economics: Modeling the behavior of economic systems.

Q: Can I use a calculator or computer software to solve quadratic equations?

Yes, you can use a calculator or computer software to solve quadratic equations. Many calculators and computer software programs have built-in functions for solving quadratic equations.

Q: What are some tips and tricks for solving quadratic equations?

Some tips and tricks for solving quadratic equations include:

  • Always start by applying the zero product property.
  • Use algebraic manipulations to simplify the equation.
  • Check your solutions by plugging them back into the original equation.
  • Use a calculator or computer software to check your solutions.

Q: Can I solve quadratic equations with complex numbers?

Yes, you can solve quadratic equations with complex numbers. Complex numbers are numbers that have both real and imaginary parts.

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

A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one.

Conclusion

In conclusion, quadratic equations are an essential part of mathematics, and they have many real-world applications. By following the steps outlined in this article, you can solve quadratic equations with ease. Remember to always apply the zero product property, simplify the equation, and check your solutions. With practice and patience, you can become proficient in solving quadratic equations and apply them to real-world problems.

Additional Resources

For more information on quadratic equations, you can check out the following resources:

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

Final Thoughts

Quadratic equations are a fundamental part of mathematics, and they have many real-world applications. By mastering the skills outlined in this article, you can become proficient in solving quadratic equations and apply them to real-world problems. Remember to always practice and review, and don't be afraid to ask for help when you need it.