Solve By Applying The Zero Product Property. 144 N 2 − 81 = 0 144n^2 - 81 = 0 144 N 2 − 81 = 0

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Introduction

The zero product property is a fundamental concept in algebra that states if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. This property is used to solve equations that involve products of variables or expressions. In this article, we will apply the zero product property to solve the quadratic equation $144n^2 - 81 = 0$.

Understanding the Zero Product Property

The zero product property can be stated as follows:

• If $ab = 0$, then $a = 0$ or $b = 0$
• If $a(b + c) = 0$, then $a = 0$ or $b + c = 0$
• If $a(b - c) = 0$, then $a = 0$ or $b - c = 0$

This property is used to solve equations that involve products of variables or expressions. By applying the zero product property, we can isolate the variable and solve for its value.

Solving the Quadratic Equation

The given quadratic equation is $144n^2 - 81 = 0$. To solve this equation, we can start by adding 81 to both sides of the equation:

$$144n^2 - 81 + 81 = 0 + 81$$

This simplifies to:

$$144n^2 = 81$$

Next, we can divide both sides of the equation by 144:

$$\frac{144n^2}{144} = \frac{81}{144}$$

This simplifies to:

$$n^2 = \frac{9}{16}$$

Applying the Zero Product Property

Now that we have isolated the variable $n^2$, we can apply the zero product property to solve for $n$. We can rewrite the equation as:

$$n^2 - \frac{9}{16} = 0$$

This can be factored as:

$$(n - \frac{3}{4})(n + \frac{3}{4}) = 0$$

Solving for $n$

Using the zero product property, we can set each factor equal to zero and solve for $n$:

$$n - \frac{3}{4} = 0 \quad \text{or} \quad n + \frac{3}{4} = 0$$

Solving for $n$ in the first equation, we get:

$$n = \frac{3}{4}$$

Solving for $n$ in the second equation, we get:

$$n = -\frac{3}{4}$$

Conclusion

In this article, we applied the zero product property to solve the quadratic equation $144n^2 - 81 = 0$. We started by adding 81 to both sides of the equation and then divided both sides by 144. We then isolated the variable $n^2$ and applied the zero product property to solve for $n$. We found two possible solutions for $n$: $n = \frac{3}{4}$ and $n = -\frac{3}{4}$.

Example Use Cases

The zero product property is a powerful tool for solving equations that involve products of variables or expressions. Here are a few example use cases:

• Solving quadratic equations: The zero product property can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$.
• Solving systems of equations: The zero product property can be used to solve systems of equations that involve products of variables or expressions.
• Solving equations with rational expressions: The zero product property can be used to solve equations that involve rational expressions.

Tips and Tricks

Here are a few tips and tricks for applying the zero product property:

• Make sure to add or subtract the same value to both sides of the equation.
• Make sure to divide both sides of the equation by the same value.
• Use the zero product property to isolate the variable.
• Check your work by plugging the solution back into the original equation.

Common Mistakes

Here are a few common mistakes to avoid when applying the zero product property:

• Failing to add or subtract the same value to both sides of the equation.
• Failing to divide both sides of the equation by the same value.
• Failing to use the zero product property to isolate the variable.
• Not checking your work by plugging the solution back into the original equation.

Conclusion

In conclusion, the zero product property is a powerful tool for solving equations that involve products of variables or expressions. By applying the zero product property, we can isolate the variable and solve for its value. We can use the zero product property to solve quadratic equations, systems of equations, and equations with rational expressions. By following the tips and tricks outlined in this article, we can avoid common mistakes and ensure that our solutions are correct.

Introduction

In our previous article, we applied the zero product property to solve the quadratic equation $144n^2 - 81 = 0$. In this article, we will answer some frequently asked questions about applying the zero product property.

Q: What is the zero product property?

A: The zero product property is a fundamental concept in algebra that states 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 isolate the variable by adding or subtracting the same value to both sides of the equation, and then dividing both sides by the same value. Once you have isolated the variable, you can set each factor equal to zero and solve for the variable.

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

A: Some common mistakes to avoid when applying the zero product property include failing to add or subtract the same value to both sides of the equation, failing to divide both sides of the equation by the same value, and failing to use the zero product property to isolate the variable.

Q: Can I use the zero product property to solve equations with rational expressions?

A: Yes, you can use the zero product property to solve equations with rational expressions. However, you need to be careful when working with rational expressions, as they can be tricky to work with.

Q: How do I check my work when applying the zero product property?

A: To check your work, you need to plug the solution back into the original equation and make sure that it is true. If the solution is not true, then you need to go back and recheck your work.

Q: Can I use the zero product property to solve systems of equations?

A: Yes, you can use the zero product property to solve systems of equations. However, you need to be careful when working with systems of equations, as they can be tricky to solve.

Q: What are some real-world applications of the zero product property?

A: The zero product property has many real-world applications, including solving quadratic equations, systems of equations, and equations with rational expressions. It is also used in many fields, such as physics, engineering, and computer science.

Q: Can I use the zero product property to solve equations with complex numbers?

A: Yes, you can use the zero product property to solve equations with complex numbers. However, you need to be careful when working with complex numbers, as they can be tricky to work with.

Q: How do I apply the zero product property to solve equations with absolute value?

A: To apply the zero product property to solve equations with absolute value, you need to isolate the variable by adding or subtracting the same value to both sides of the equation, and then dividing both sides by the same value. Once you have isolated the variable, you can set each factor equal to zero and solve for the variable.

Q: Can I use the zero product property to solve equations with exponents?

A: Yes, you can use the zero product property to solve equations with exponents. However, you need to be careful when working with exponents, as they can be tricky to work with.

Q: How do I apply the zero product property to solve equations with fractions?

A: To apply the zero product property to solve equations with fractions, you need to isolate the variable by adding or subtracting the same value to both sides of the equation, and then dividing both sides by the same value. Once you have isolated the variable, you can set each factor equal to zero and solve for the variable.

Conclusion

In conclusion, the zero product property is a powerful tool for solving equations that involve products of variables or expressions. By applying the zero product property, we can isolate the variable and solve for its value. We can use the zero product property to solve quadratic equations, systems of equations, and equations with rational expressions. By following the tips and tricks outlined in this article, we can avoid common mistakes and ensure that our solutions are correct.

Example Use Cases

Here are a few example use cases for the zero product property:

• Solving quadratic equations: The zero product property can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$.
• Solving systems of equations: The zero product property can be used to solve systems of equations that involve products of variables or expressions.
• Solving equations with rational expressions: The zero product property can be used to solve equations that involve rational expressions.

Tips and Tricks

Here are a few tips and tricks for applying the zero product property:

• Make sure to add or subtract the same value to both sides of the equation.
• Make sure to divide both sides of the equation by the same value.
• Use the zero product property to isolate the variable.
• Check your work by plugging the solution back into the original equation.

Common Mistakes

Here are a few common mistakes to avoid when applying the zero product property:

• Failing to add or subtract the same value to both sides of the equation.
• Failing to divide both sides of the equation by the same value.
• Failing to use the zero product property to isolate the variable.
• Not checking your work by plugging the solution back into the original equation.