If $(x+3)(x-4)=0$, Which Of The Following Is A Possible Value Of $x$?

Mar 13, 2025 by ADMIN 74 views

If $(x+3)(x-4)=0$, Which of the Following is a Possible Value of $x$?

Introduction

In algebra, solving equations is a fundamental concept that helps us find the values of unknown variables. One of the most common types of equations is the quadratic equation, which can be factored into the product of two binomials. In this article, we will explore how to solve a quadratic equation of the form $(x+3)(x-4)=0$ and determine which of the following is a possible value of $x$.

Understanding the Equation

The given equation is a quadratic equation in the form of $(x+3)(x-4)=0$. To solve this equation, we need to find the values of $x$ that make the equation true. In other words, we need to find the values of $x$ that make the product of the two binomials equal to zero.

Solving the Equation

To solve the equation, we can use the zero-product property, which states that if the product of two numbers is zero, then at least one of the numbers must be zero. In this case, we can set each binomial equal to zero and solve for $x$.

Setting the First Binomial Equal to Zero

Let's start by setting the first binomial, $(x+3)$, equal to zero.

x+3=0

Subtracting 3 from both sides gives us:

x=-3

Setting the Second Binomial Equal to Zero

Now, let's set the second binomial, $(x-4)$, equal to zero.

x-4=0

Adding 4 to both sides gives us:

x=4

Possible Values of $x$

Now that we have solved the equation, we can see that there are two possible values of $x$: $x=-3$ and $x=4$. These values make the equation true, and therefore, they are possible values of $x$.

Conclusion

In conclusion, solving the quadratic equation $(x+3)(x-4)=0$ involves using the zero-product property to find the values of $x$ that make the equation true. By setting each binomial equal to zero and solving for $x$, we found that the possible values of $x$ are $x=-3$ and $x=4$. These values are the solutions to the equation, and they provide a possible value of $x$.

Frequently Asked Questions

Q: What is the zero-product property?

A: The zero-product property states that if the product of two numbers is zero, then at least one of the numbers must be zero.

Q: How do I solve a quadratic equation of the form $(x+3)(x-4)=0$?

A: To solve a quadratic equation of the form $(x+3)(x-4)=0$, you can use the zero-product property to set each binomial equal to zero and solve for $x$.

Q: What are the possible values of $x$ for the equation $(x+3)(x-4)=0$?

A: The possible values of $x$ for the equation $(x+3)(x-4)=0$ are $x=-3$ and $x=4$.

Final Thoughts

Solving quadratic equations is an essential skill in algebra, and it has many real-world applications. By understanding how to solve equations of the form $(x+3)(x-4)=0$, you can apply this knowledge to a wide range of problems. Whether you're working with quadratic equations or other types of equations, the concepts and techniques you learn in this article will help you become a more confident and proficient problem-solver.

Additional Resources

Q&A: Solving Quadratic Equations

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It can be written in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use various methods such as factoring, the quadratic formula, or completing the square. The method you choose depends on the form of the equation and the complexity of the solutions.

Q: What is the zero-product property?

A: The zero-product property states that if the product of two numbers is zero, then at least one of the numbers must be zero. This property is used to solve quadratic equations by setting each binomial equal to zero and solving for $x$.

Q: How do I use the zero-product property to solve a quadratic equation?

A: To use the zero-product property, you need to set each binomial equal to zero and solve for $x$. For example, if you have the equation $(x+3)(x-4)=0$, you can set each binomial equal to zero and solve for $x$:

x+3=0

x-4=0

Solving for $x$ gives you $x=-3$ and $x=4$.

Q: What are the possible values of $x$ for the equation $(x+3)(x-4)=0$?

A: The possible values of $x$ for the equation $(x+3)(x-4)=0$ are $x=-3$ and $x=4$.

Q: How do I determine which of the following is a possible value of $x$?

A: To determine which of the following is a possible value of $x$, you need to check if the value satisfies the equation. In this case, we have the equation $(x+3)(x-4)=0$, and we need to check if $x=-3$ or $x=4$ satisfies the equation.

Q: What is the significance of solving quadratic equations?

A: Solving quadratic equations is significant because it helps us find the values of unknown variables. Quadratic equations have many real-world applications, such as modeling the trajectory of a projectile, the motion of an object under constant acceleration, or the growth of a population.

Q: How do I apply the concepts of solving quadratic equations to real-world problems?

A: To apply the concepts of solving quadratic equations to real-world problems, you need to identify the variables and the relationships between them. You can then use the methods of solving quadratic equations to find the values of the variables that satisfy the equation.

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

A: Some common mistakes to avoid when solving quadratic equations include:

Not setting each binomial equal to zero
Not solving for $x$ correctly
Not checking if the solutions satisfy the equation
Not using the correct method for solving the equation

Q: How do I check if the solutions satisfy the equation?

A: To check if the solutions satisfy the equation, you need to plug the solutions back into the original equation and check if it is true. For example, if you have the equation $(x+3)(x-4)=0$ and you find that $x=-3$ and $x=4$ are the solutions, you can plug these values back into the equation to check if it is true.

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

A: Some real-world applications of solving quadratic equations include:

Modeling the trajectory of a projectile
The motion of an object under constant acceleration
The growth of a population
The design of electrical circuits
The analysis of data in statistics

Q: How do I use technology to solve quadratic equations?

A: You can use technology such as calculators or computer software to solve quadratic equations. These tools can help you find the solutions to the equation and check if they satisfy the equation.

Q: What are some common types of quadratic equations?

A: Some common types of quadratic equations include:

Monic quadratic equations: $x^2 + bx + c = 0$
Non-monic quadratic equations: $ax^2 + bx + c = 0$
Quadratic equations with complex coefficients: $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are complex numbers.

Q: How do I solve quadratic equations with complex coefficients?

A: To solve quadratic equations with complex coefficients, you need to use the quadratic formula and the properties of complex numbers. The quadratic formula is:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

You can use this formula to find the solutions to the equation, and then check if they satisfy the equation.

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

A: Some common mistakes to avoid when solving quadratic equations with complex coefficients include:

Not using the correct formula for solving the equation
Not checking if the solutions satisfy the equation
Not using the correct method for solving the equation

Q: How do I check if the solutions satisfy the equation when solving quadratic equations with complex coefficients?

A: To check if the solutions satisfy the equation when solving quadratic equations with complex coefficients, you need to plug the solutions back into the original equation and check if it is true. You can use the properties of complex numbers to simplify the equation and check if it is true.

Q: What are some real-world applications of solving quadratic equations with complex coefficients?

A: Some real-world applications of solving quadratic equations with complex coefficients include:

Modeling the behavior of electrical circuits
The analysis of data in statistics
The design of electronic devices
The study of quantum mechanics

Q: How do I use technology to solve quadratic equations with complex coefficients?

A: You can use technology such as calculators or computer software to solve quadratic equations with complex coefficients. These tools can help you find the solutions to the equation and check if they satisfy the equation.

Q: What are some common types of quadratic equations with complex coefficients?

A: Some common types of quadratic equations with complex coefficients include:

Quadratic equations with complex roots: $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are complex numbers and the roots are complex numbers.
Quadratic equations with complex coefficients and real roots: $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are complex numbers and the roots are real numbers.

Q: How do I solve quadratic equations with complex coefficients and real roots?

A: To solve quadratic equations with complex coefficients and real roots, you need to use the quadratic formula and the properties of complex numbers. The quadratic formula is:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

You can use this formula to find the solutions to the equation, and then check if they satisfy the equation.

Q: What are some common mistakes to avoid when solving quadratic equations with complex coefficients and real roots?

A: Some common mistakes to avoid when solving quadratic equations with complex coefficients and real roots include:

Not using the correct formula for solving the equation
Not checking if the solutions satisfy the equation
Not using the correct method for solving the equation

Q: How do I check if the solutions satisfy the equation when solving quadratic equations with complex coefficients and real roots?

A: To check if the solutions satisfy the equation when solving quadratic equations with complex coefficients and real roots, you need to plug the solutions back into the original equation and check if it is true. You can use the properties of complex numbers to simplify the equation and check if it is true.

Q: What are some real-world applications of solving quadratic equations with complex coefficients and real roots?

A: Some real-world applications of solving quadratic equations with complex coefficients and real roots include:

Modeling the behavior of electrical circuits
The analysis of data in statistics
The design of electronic devices
The study of quantum mechanics

Q: How do I use technology to solve quadratic equations with complex coefficients and real roots?

A: You can use technology such as calculators or computer software to solve quadratic equations with complex coefficients and real roots. These tools can help you find the solutions to the equation and check if they satisfy the equation.

Q: What are some common types of quadratic equations with complex coefficients and real roots?

A: Some common types of quadratic equations with complex coefficients and real roots include:

Quadratic equations with complex coefficients and real roots: $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are complex numbers and the roots are real numbers.
Quadratic equations with complex coefficients and real roots and complex coefficients: $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are complex numbers and the roots are real numbers and complex numbers.

Q: How do I solve quadratic equations with complex coefficients and real roots and complex coefficients?

A: To solve quadratic equations with complex coefficients and real roots and complex coefficients, you need to use the quadratic formula and the properties of complex