Solve The Equation: \[$(x-3)(5x+6)=0\$\]

Feb 26, 2025 by ADMIN 41 views

Introduction

In mathematics, equations are a fundamental concept that help us solve problems and understand various mathematical concepts. One of the most common types of equations is the quadratic equation, which is a polynomial equation of degree two. In this article, we will focus on solving a quadratic equation of the form $(x-3)(5x+6)=0$ . This equation is a product of two binomials, and we will use the zero-product property to solve for the variable $x$ .

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 $ab=0$ , then either $a=0$ or $b=0$ . This property is a fundamental concept in algebra and is used to solve equations of the form $(x-a)(x-b)=0$ .

Applying the Zero-Product Property to the Equation

To solve the equation $(x-3)(5x+6)=0$ , we can apply the zero-product property by setting each factor equal to zero. This gives us two separate equations:

$x-3=0$ and $5x+6=0$

Solving the First Equation

To solve the first equation, $x-3=0$ , we can add 3 to both sides of the equation. This gives us:

$x-3+3=0+3$

$x=3$

Solving the Second Equation

To solve the second equation, $5x+6=0$ , we can subtract 6 from both sides of the equation. This gives us:

$5x+6-6=0-6$

$5x=-6$

Next, we can divide both sides of the equation by 5 to solve for $x$ . This gives us:

$\frac{5x}{5}=\frac{-6}{5}$

$x=-\frac{6}{5}$

Conclusion

In conclusion, we have solved the equation $(x-3)(5x+6)=0$ using the zero-product property. We found that the solutions to the equation are $x=3$ and $x=-\frac{6}{5}$ . These solutions are the values of $x$ that make the equation true.

Applications of the Zero-Product Property

The zero-product property has many applications in mathematics and science. For example, it is used to solve equations in physics, engineering, and computer science. It is also used to find the roots of a polynomial equation, which is a fundamental concept in algebra.

Real-World Examples of the Zero-Product Property

The zero-product property has many real-world applications. For example, it is used to solve problems in physics, such as finding the time it takes for an object to fall to the ground. It is also used to solve problems in engineering, such as designing a bridge or a building.

Tips for Solving Equations Using the Zero-Product Property

Here are some tips for solving equations using the zero-product property:

Make sure to set each factor equal to zero.
Solve each equation separately.
Check your solutions by plugging them back into the original equation.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving equations using the zero-product property:

Not setting each factor equal to zero.
Not solving each equation separately.
Not checking your solutions by plugging them back into the original equation.

Conclusion

In conclusion, the zero-product property is a fundamental concept in algebra that is used to solve equations of the form $(x-a)(x-b)=0$ . It is a powerful tool that has many applications in mathematics and science. By following the tips and avoiding the common mistakes, you can become proficient in solving equations using the zero-product property.

Final Thoughts

The zero-product property is a simple yet powerful concept that can be used to solve a wide range of equations. It is a fundamental concept in algebra that is used to find the roots of a polynomial equation. By mastering the zero-product property, you can become proficient in solving equations and apply it to real-world problems.

References

Introduction

In our previous article, we solved the equation $(x-3)(5x+6)=0$ using the zero-product property. In this article, we will answer some common questions that students often have when solving equations of this form.

Q: What is the zero-product property?

A: The zero-product property is a fundamental concept in algebra 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. In other words, if $ab=0$ , then either $a=0$ or $b=0$ .

Q: How do I apply the zero-product property to solve an equation?

A: To apply the zero-product property, you need to set each factor equal to zero and solve for the variable. For example, if you have the equation $(x-3)(5x+6)=0$ , you would set each factor equal to zero and solve for $x$ .

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

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

Not setting each factor equal to zero.
Not solving each equation separately.
Not checking your solutions by plugging them back into the original equation.

Q: How do I check my solutions?

A: To check your solutions, you need to plug each solution back into the original equation and make sure that it is true. For example, if you have the equation $(x-3)(5x+6)=0$ and you found that $x=3$ is a solution, you would plug $x=3$ back into the equation and make sure that it is true.

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

A: The zero-product property has many real-world applications, including:

Solving problems in physics, such as finding the time it takes for an object to fall to the ground.
Solving problems in engineering, such as designing a bridge or a building.
Solving problems in computer science, such as finding the roots of a polynomial equation.

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

A: To use the zero-product property to solve quadratic equations, you need to set each factor equal to zero and solve for the variable. For example, if you have the equation $(x-3)(5x+6)=0$ , you would set each factor equal to zero and solve for $x$ .

Q: What are some tips for solving equations using the zero-product property?

A: Some tips for solving equations using the zero-product property include:

Make sure to set each factor equal to zero.
Solve each equation separately.
Check your solutions by plugging them back into the original equation.

Q: How do I know if I have found all the solutions to an equation?

A: To know if you have found all the solutions to an equation, you need to check your solutions by plugging them back into the original equation and make sure that they are true. If you have found all the solutions, then you can be confident that you have solved the equation.

Q: What are some common mistakes to avoid when checking solutions?

A: Some common mistakes to avoid when checking solutions include:

Not plugging each solution back into the original equation.
Not making sure that each solution is true.
Not checking for extraneous solutions.

Conclusion

In conclusion, the zero-product property is a fundamental concept in algebra that is used to solve equations of the form $(x-a)(x-b)=0$ . By following the tips and avoiding the common mistakes, you can become proficient in solving equations using the zero-product property.

Solve The Equation: \[$(x-3)(5x+6)=0\$\]

Introduction

Understanding the Zero-Product Property

Applying the Zero-Product Property to the Equation

Solving the First Equation

Solving the Second Equation

Conclusion

Applications of the Zero-Product Property

Real-World Examples of the Zero-Product Property

Tips for Solving Equations Using the Zero-Product Property

Common Mistakes to Avoid

Conclusion

Final Thoughts

References

Further Reading

Introduction

Q: What is the zero-product property?

Q: How do I apply the zero-product property to solve an equation?

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

Q: How do I check my solutions?

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

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

Q: What are some tips for solving equations using the zero-product property?

Q: How do I know if I have found all the solutions to an equation?

Q: What are some common mistakes to avoid when checking solutions?

Conclusion

Final Thoughts

References

Further Reading