Solve The Equation:3) ( R + 5 ) ( R − 3 ) = 0 (r+5)(r-3)=0 ( R + 5 ) ( R − 3 ) = 0

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Introduction

In mathematics, equations are a fundamental concept that help us understand and describe various phenomena. Solving equations is a crucial skill that is essential in mathematics, science, and engineering. In this article, we will focus on solving a quadratic equation of the form $(r+5)(r-3)=0$. We will use algebraic methods to find the solutions to this equation.

Understanding the Equation

The given equation is a quadratic equation in the form of a product of two binomials. It can be written as:

$$(r+5)(r-3)=0$$

This equation can be expanded using the distributive property, but in this case, we can use the zero-product property to find the solutions.

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 $b=0$. We can use this property to solve the given equation.

Solving the Equation

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

$$(r+5)=0 \quad \text{or} \quad (r-3)=0$$

Solving the first equation, we get:

$$r+5=0 \quad \Rightarrow \quad r=-5$$

Solving the second equation, we get:

$$r-3=0 \quad \Rightarrow \quad r=3$$

Therefore, the solutions to the equation $(r+5)(r-3)=0$ are $r=-5$ and $r=3$.

Checking the Solutions

To verify that these solutions are correct, we can substitute them back into the original equation:

$$(r+5)(r-3)=0$$

Substituting $r=-5$, we get:

$((-5)+5)((-5)-3)=0 \quad \Rightarrow \quad (0)(-8)=0 \quad \Rightarrow \quad 0=0$$

This shows that $r=-5$ is a solution to the equation.

Substituting $r=3$, we get:

$((3)+5)((3)-3)=0 \quad \Rightarrow \quad (8)(0)=0 \quad \Rightarrow \quad 0=0$$

This shows that $r=3$ is also a solution to the equation.

Conclusion

In this article, we solved the quadratic equation $(r+5)(r-3)=0$ using the zero-product property. We found that the solutions to this equation are $r=-5$ and $r=3$. We also verified that these solutions are correct by substituting them back into the original equation.

Applications of Solving Equations

Solving equations is a fundamental skill that has numerous applications in mathematics, science, and engineering. Some of the applications of solving equations include:

• Physics: Solving equations is used to describe the motion of objects, including the position, velocity, and acceleration of objects.
• Engineering: Solving equations is used to design and optimize systems, including electrical circuits, mechanical systems, and thermal systems.
• Computer Science: Solving equations is used in computer graphics, game development, and scientific simulations.
• Economics: Solving equations is used to model economic systems, including the behavior of markets and the impact of policy changes.

Tips for Solving Equations

Here are some tips for solving equations:

• Read the problem carefully: Before starting to solve the equation, read the problem carefully to understand what is being asked.
• Use algebraic methods: Algebraic methods, such as the zero-product property, can be used to solve equations.
• Check the solutions: Once you have found the solutions, check them by substituting them back into the original equation.
• Practice, practice, practice: Solving equations is a skill that requires practice to develop.

Conclusion

In conclusion, solving equations is a fundamental skill that is essential in mathematics, science, and engineering. In this article, we solved the quadratic equation $(r+5)(r-3)=0$ using the zero-product property. We found that the solutions to this equation are $r=-5$ and $r=3$. We also verified that these solutions are correct by substituting them back into the original equation.

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Introduction

In our previous article, we solved the quadratic equation $(r+5)(r-3)=0$ using the zero-product property. We found that the solutions to this equation are $r=-5$ and $r=3$. In this article, we will answer some frequently asked questions about solving equations.

Q&A

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 other words, if $ab=0$, then either $a=0$ or $b=0$.

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

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

$$(r+5)=0 \quad \text{or} \quad (r-3)=0$$

Solving the first equation, you get:

$$r+5=0 \quad \Rightarrow \quad r=-5$$

Solving the second equation, you get:

$$r-3=0 \quad \Rightarrow \quad r=3$$

Q: What if I have a quadratic equation in the form of $ax^2+bx+c=0$? How do I solve it?

A: To solve a quadratic equation in the form of $ax^2+bx+c=0$, you can use the quadratic formula:

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

This formula will give you two solutions for the equation.

Q: What if I have a system of equations? How do I solve it?

A: To solve a system of equations, you need to find the values of the variables that satisfy all the equations in the system. There are several methods to solve a system of equations, including substitution, elimination, and graphing.

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

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

• Not reading the problem carefully: Before starting to solve the equation, read the problem carefully to understand what is being asked.
• Not using the correct algebraic methods: Make sure to use the correct algebraic methods to solve the equation.
• Not checking the solutions: Once you have found the solutions, check them by substituting them back into the original equation.
• Not practicing, practicing, practicing: Solving equations is a skill that requires practice to develop.

Conclusion

In conclusion, solving equations is a fundamental skill that is essential in mathematics, science, and engineering. In this article, we answered some frequently asked questions about solving equations. We hope that this article has been helpful in clarifying some of the concepts and methods involved in solving equations.

Additional Resources

If you are interested in learning more about solving equations, here are some additional resources that you may find helpful:

• Textbooks: There are many textbooks available that cover the topic of solving equations, including "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart.
• Online resources: There are many online resources available that provide tutorials and examples on solving equations, including Khan Academy and Mathway.
• Practice problems: Practice problems are an essential part of learning to solve equations. You can find practice problems in textbooks, online resources, or by creating your own problems.

Final Tips

Here are some final tips for solving equations:

• Practice, practice, practice: Solving equations is a skill that requires practice to develop.
• Read the problem carefully: Before starting to solve the equation, read the problem carefully to understand what is being asked.
• Use algebraic methods: Algebraic methods, such as the zero-product property, can be used to solve equations.
• Check the solutions: Once you have found the solutions, check them by substituting them back into the original equation.