Solve For X: { (2x - 5)(3x + 1) = 0$}$

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Introduction

In algebra, solving for x is a fundamental concept that involves finding the value of a variable that satisfies an equation. The equation {(2x - 5)(3x + 1) = 0$}$ is a quadratic equation that can be solved using the zero product property. In this article, we will explore the steps to solve for x in this equation and provide a detailed explanation of the solution process.

Understanding 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 the equation {(2x - 5)(3x + 1) = 0$}$, we have two factors: ${2x - 5\$} and ${3x + 1\$}. According to the zero product property, if the product of these two factors is equal to zero, then either ${2x - 5 = 0\$} or ${3x + 1 = 0\$}.

Solving for x in the First Factor

Let's start by solving for x in the first factor: ${2x - 5 = 0\$}. To solve for x, we need to isolate the variable x on one side of the equation. We can do this by adding 5 to both sides of the equation:

${2x - 5 + 5 = 0 + 5\$}

This simplifies to:

${2x = 5\$}

Next, we can divide both sides of the equation by 2 to solve for x:

{\frac{2x}{2} = \frac{5}{2}$}$

This simplifies to:

{x = \frac{5}{2}$}$

Solving for x in the Second Factor

Now, let's solve for x in the second factor: ${3x + 1 = 0\$}. To solve for x, we need to isolate the variable x on one side of the equation. We can do this by subtracting 1 from both sides of the equation:

${3x + 1 - 1 = 0 - 1\$}

This simplifies to:

${3x = -1\$}

Next, we can divide both sides of the equation by 3 to solve for x:

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

This simplifies to:

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

Conclusion

In this article, we have solved for x in the equation {(2x - 5)(3x + 1) = 0$}$ using the zero product property. We have found two possible values for x: {x = \frac{5}{2}$}$ and {x = -\frac{1}{3}$}$. These values satisfy the equation and are the solutions to the problem.

Real-World Applications

The concept of solving for x is used in a wide range of real-world applications, including:

  • Physics: Solving for x is used to calculate the position and velocity of objects in motion.
  • Engineering: Solving for x is used to design and optimize systems, such as bridges and buildings.
  • Computer Science: Solving for x is used in algorithms and data structures to solve complex problems.

Tips and Tricks

Here are some tips and tricks to help you solve for x:

  • Use the zero product property: The zero product property is a powerful tool for solving equations. Use it to your advantage by factoring the equation and setting each factor equal to zero.
  • Simplify the equation: Simplify the equation by combining like terms and eliminating any unnecessary variables.
  • Check your work: Always check your work by plugging the solution back into the original equation.

Common Mistakes

Here are some common mistakes to avoid when solving for x:

  • Not using the zero product property: Failing to use the zero product property can lead to incorrect solutions.
  • Not simplifying the equation: Failing to simplify the equation can lead to incorrect solutions.
  • Not checking your work: Failing to check your work can lead to incorrect solutions.

Conclusion

Q: What is the zero product property?

A: 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 the equation {(2x - 5)(3x + 1) = 0$}$, we have two factors: ${2x - 5\$} and ${3x + 1\$}. According to the zero product property, if the product of these two factors is equal to zero, then either ${2x - 5 = 0\$} or ${3x + 1 = 0\$}.

Q: How do I solve for x in the first factor?

A: To solve for x in the first factor, we need to isolate the variable x on one side of the equation. We can do this by adding 5 to both sides of the equation:

${2x - 5 + 5 = 0 + 5\$}

This simplifies to:

${2x = 5\$}

Next, we can divide both sides of the equation by 2 to solve for x:

{\frac{2x}{2} = \frac{5}{2}$}$

This simplifies to:

{x = \frac{5}{2}$}$

Q: How do I solve for x in the second factor?

A: To solve for x in the second factor, we need to isolate the variable x on one side of the equation. We can do this by subtracting 1 from both sides of the equation:

${3x + 1 - 1 = 0 - 1\$}

This simplifies to:

${3x = -1\$}

Next, we can divide both sides of the equation by 3 to solve for x:

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

This simplifies to:

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

Q: What are some common mistakes to avoid when solving for x?

A: Here are some common mistakes to avoid when solving for x:

  • Not using the zero product property: Failing to use the zero product property can lead to incorrect solutions.
  • Not simplifying the equation: Failing to simplify the equation can lead to incorrect solutions.
  • Not checking your work: Failing to check your work can lead to incorrect solutions.

Q: How do I check my work when solving for x?

A: To check your work, plug the solution back into the original equation. If the solution satisfies the equation, then it is the correct solution. If the solution does not satisfy the equation, then it is not the correct solution.

Q: What are some real-world applications of solving for x?

A: The concept of solving for x is used in a wide range of real-world applications, including:

  • Physics: Solving for x is used to calculate the position and velocity of objects in motion.
  • Engineering: Solving for x is used to design and optimize systems, such as bridges and buildings.
  • Computer Science: Solving for x is used in algorithms and data structures to solve complex problems.

Q: How do I simplify the equation when solving for x?

A: To simplify the equation, combine like terms and eliminate any unnecessary variables. For example, in the equation ${2x - 5 = 0\$}, we can combine the like terms ${2x\$} and {-5$}$ to get:

${2x - 5 = 0\$}

This simplifies to:

${2x = 5\$}

Next, we can divide both sides of the equation by 2 to solve for x:

{\frac{2x}{2} = \frac{5}{2}$}$

This simplifies to:

{x = \frac{5}{2}$}$

Q: What are some tips and tricks for solving for x?

A: Here are some tips and tricks for solving for x:

  • Use the zero product property: The zero product property is a powerful tool for solving equations. Use it to your advantage by factoring the equation and setting each factor equal to zero.
  • Simplify the equation: Simplify the equation by combining like terms and eliminating any unnecessary variables.
  • Check your work: Always check your work by plugging the solution back into the original equation.

Conclusion

Solving for x is a fundamental concept in algebra that involves finding the value of a variable that satisfies an equation. By following the steps outlined in this article, you can solve for x and find the solutions to the problem. Remember to use the zero product property, simplify the equation, and check your work to ensure that you are getting the correct solutions.