Solve For { Z $} . . . { (z-6)(z-7) = 0\$}

by ADMIN 43 views

Introduction

When solving equations, we often come across expressions that involve the product of two or more factors. In this case, we have a quadratic equation in the form of {(z-6)(z-7) = 0$}$. Our goal is to find the value of { z $}$ that satisfies this equation. To do this, we will use the concept of factoring and the zero-product property.

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 other words, if {(z-6)(z-7) = 0$}$, then either {(z-6) = 0$}$ or {(z-7) = 0$}$.

Solving for { z $}$

To solve for { z $}$, we need to isolate the variable { z $}$ in each of the two equations.

Solving {(z-6) = 0$}$

We can start by adding 6 to both sides of the equation:

{z-6+6=0+6$}$

This simplifies to:

{z=6$}$

So, one possible value of { z $}$ is 6.

Solving {(z-7) = 0$}$

Similarly, we can add 7 to both sides of the equation:

{z-7+7=0+7$}$

This simplifies to:

{z=7$}$

So, another possible value of { z $}$ is 7.

Conclusion

In this discussion, we used the zero-product property to solve the equation {(z-6)(z-7) = 0$}$. We found that the possible values of { z $}$ are 6 and 7. These values satisfy the equation, and we can verify this by plugging them back into the original equation.

Example Use Cases

The concept of factoring and the zero-product property is used extensively in mathematics, particularly in algebra and calculus. Here are a few example use cases:

  • Solving quadratic equations: When solving quadratic equations, we often need to factor the expression and use the zero-product property to find the values of the variable.
  • Graphing functions: When graphing functions, we need to find the x-intercepts, which are the values of the variable that make the function equal to zero. The zero-product property is used to find these x-intercepts.
  • Solving systems of equations: When solving systems of equations, we need to find the values of the variables that satisfy all the equations. The zero-product property is used to find these values.

Tips and Tricks

Here are a few tips and tricks to keep in mind when using the zero-product property:

  • Make sure to factor the expression correctly: When factoring the expression, make sure to factor it correctly and identify the factors that are equal to zero.
  • Use the zero-product property carefully: When using the zero-product property, make sure to identify the factors that are equal to zero and solve for the variable correctly.
  • Verify the solutions: When solving the equation, make sure to verify the solutions by plugging them back into the original equation.

Conclusion

In conclusion, the zero-product property is a powerful tool in mathematics that allows us to solve equations by identifying the factors that are equal to zero. By using this property, we can solve quadratic equations, graph functions, and solve systems of equations. Remember to factor the expression correctly, use the zero-product property carefully, and verify the solutions to ensure that you are getting the correct answers.

Introduction

In our previous discussion, we explored the concept of the zero-product property and how it can be used to solve equations. In this article, we will answer some frequently asked questions about solving equations with the zero-product property.

Q: What is the zero-product property?

A: The zero-product property is a mathematical concept that 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 solve an equation?

A: To apply the zero-product property, you need to factor the expression and identify the factors that are equal to zero. Then, you can solve for the variable by setting each factor equal to zero and solving for the variable.

Q: What are some common mistakes to avoid when using the zero-product property?

A: Some common mistakes to avoid when using the zero-product property include:

  • Not factoring the expression correctly: Make sure to factor the expression correctly and identify the factors that are equal to zero.
  • Not using the zero-product property carefully: Make sure to identify the factors that are equal to zero and solve for the variable correctly.
  • Not verifying the solutions: Make sure to verify the solutions by plugging them back into the original equation.

Q: Can I use the zero-product property to solve equations with more than two factors?

A: Yes, you can use the zero-product property to solve equations with more than two factors. However, you need to factor the expression correctly and identify the factors that are equal to zero.

Q: How do I know if I have factored the expression correctly?

A: To check if you have factored the expression correctly, you can use the following steps:

  • Check if the factors are equal to zero: Make sure that the factors you have identified are equal to zero.
  • Check if the expression is equal to zero: Make sure that the expression is equal to zero when you substitute the values of the factors into the original equation.

Q: Can I use the zero-product property to solve equations with variables on both sides?

A: Yes, you can use the zero-product property to solve equations with variables on both sides. However, you need to isolate the variable on one side of the equation before applying the zero-product property.

Q: How do I know if I have solved the equation correctly?

A: To check if you have solved the equation correctly, you can use the following steps:

  • Verify the solutions: Make sure to verify the solutions by plugging them back into the original equation.
  • Check if the equation is true: Make sure that the equation is true when you substitute the values of the variable into the original equation.

Q: Can I use the zero-product property to solve equations with fractions or decimals?

A: Yes, you can use the zero-product property to solve equations with fractions or decimals. However, you need to simplify the expression and identify the factors that are equal to zero.

Q: How do I simplify an expression with fractions or decimals?

A: To simplify an expression with fractions or decimals, you can use the following steps:

  • Simplify the fractions or decimals: Simplify the fractions or decimals by dividing or multiplying them by a common factor.
  • Identify the factors that are equal to zero: Identify the factors that are equal to zero and solve for the variable.

Conclusion

In conclusion, the zero-product property is a powerful tool in mathematics that allows us to solve equations by identifying the factors that are equal to zero. By using this property, we can solve quadratic equations, graph functions, and solve systems of equations. Remember to factor the expression correctly, use the zero-product property carefully, and verify the solutions to ensure that you are getting the correct answers.