Solving (x+3)(x+17)=0 Step-by-Step Guide To Quadratic Equations

Jul 11, 2025 by ADMIN 64 views

Solving the Quadratic Equation (x+3)(x+17)=0

In mathematics, one of the fundamental tasks is to solve equations, and quadratic equations form a significant part of this domain. A quadratic equation is a polynomial equation of the second degree, meaning it has the highest power of the variable as 2. These equations are prevalent in various fields, including physics, engineering, economics, and computer science, making their solutions crucial for numerous applications. This article delves into the process of solving the specific quadratic equation (x+3)(x+17)=0, providing a step-by-step guide and exploring the underlying mathematical principles.

Understanding Quadratic Equations

Before we dive into the solution, it's essential to understand the basic form and properties of quadratic equations. A general quadratic equation can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to 0. The solutions to this equation, also known as roots or zeros, are the values of x that satisfy the equation. These roots can be real or complex numbers, and a quadratic equation typically has two roots.

There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. The choice of method often depends on the specific form of the equation and the ease with which it can be manipulated. In the given equation, (x+3)(x+17)=0, the equation is already factored, which simplifies the solving process significantly.

The Zero Product Property

The key to solving the equation (x+3)(x+17)=0 lies in the Zero Product Property. This property states that if the product of two or more factors is equal to zero, then at least one of the factors must be zero. In other words, if AB = 0, then either A = 0 or B = 0 (or both). This property is a cornerstone of algebra and is particularly useful in solving factored equations.

Applying the Zero Product Property to our equation, we recognize that the equation is already in a factored form. The two factors are (x+3) and (x+17). According to the Zero Product Property, for the product of these two factors to be zero, at least one of them must be zero. This leads us to two separate equations:

x + 3 = 0
x + 17 = 0

Solving for x

Now, we have two simple linear equations that we can solve individually to find the values of x. These equations are much easier to handle than the original quadratic equation, thanks to the factored form and the Zero Product Property.

Solving x + 3 = 0

To solve the equation x + 3 = 0, we need to isolate x on one side of the equation. This can be achieved by subtracting 3 from both sides of the equation:

x + 3 - 3 = 0 - 3

This simplifies to:

x = -3

So, one of the solutions to the equation (x+3)(x+17)=0 is x = -3. This means that when x is -3, the expression (x+3) becomes zero, and consequently, the entire product (x+3)(x+17) becomes zero.

Solving x + 17 = 0

Similarly, to solve the equation x + 17 = 0, we need to isolate x. We can do this by subtracting 17 from both sides of the equation:

x + 17 - 17 = 0 - 17

This simplifies to:

x = -17

Thus, the other solution to the equation (x+3)(x+17)=0 is x = -17. When x is -17, the expression (x+17) becomes zero, making the entire product (x+3)(x+17) equal to zero.

Solutions to the Quadratic Equation

We have found two values of x that satisfy the equation (x+3)(x+17)=0: x = -3 and x = -17. These are the roots of the quadratic equation. We can verify these solutions by substituting them back into the original equation.

Verification for x = -3

Substituting x = -3 into the equation (x+3)(x+17)=0, we get:

(-3 + 3)(-3 + 17) = 0

(0)(14) = 0

0 = 0

This confirms that x = -3 is a valid solution.

Verification for x = -17

Substituting x = -17 into the equation (x+3)(x+17)=0, we get:

(-17 + 3)(-17 + 17) = 0

(-14)(0) = 0

0 = 0

This verifies that x = -17 is also a valid solution.

Expressing the Solutions

The solutions to the equation (x+3)(x+17)=0 are x = -3 and x = -17. We can express these solutions as a set: {-3, -17}. This set represents all the values of x that make the equation true. In many contexts, it is important to present the solutions clearly and concisely.

Alternative Methods and Insights

While factoring and the Zero Product Property provide a straightforward solution in this case, it's worth noting that other methods can also be used to solve quadratic equations. These include:

Expanding and using the quadratic formula: We could expand the given equation to the standard form ax^2 + bx + c = 0 and then apply the quadratic formula. However, this method is less efficient when the equation is already factored.
Completing the square: This method involves manipulating the equation to form a perfect square trinomial. It is a more general method but can be more complex for equations that are easily factored.

In this specific case, the factored form of the equation makes the Zero Product Property the most efficient method. It highlights the importance of recognizing the structure of an equation to choose the most appropriate solution technique.

Real-World Applications

Understanding how to solve quadratic equations is not just an academic exercise; it has numerous real-world applications. Quadratic equations appear in various fields:

Physics: Projectile motion, where the height of a projectile can be modeled by a quadratic equation.
Engineering: Designing structures and systems, such as bridges and electrical circuits.
Economics: Modeling supply and demand curves, as well as cost and revenue functions.
Computer Science: Optimization problems and algorithm design.

The ability to solve quadratic equations is, therefore, a valuable skill in many professions.

Conclusion

Solving the quadratic equation (x+3)(x+17)=0 demonstrates the power and simplicity of the Zero Product Property when dealing with factored equations. The solutions, x = -3 and x = -17, are found by setting each factor to zero and solving the resulting linear equations. This method is efficient and provides a clear understanding of the roots of the equation. Quadratic equations are fundamental in mathematics and have widespread applications in various fields, making the ability to solve them an essential skill. By mastering these techniques, students and professionals alike can tackle a wide range of problems with confidence.

Solve the equation (x+3)(x+17)=0

The task is to solve the equation (x+3)(x+17)=0. This equation is a quadratic equation presented in a factored form, which makes it relatively straightforward to solve. The key to solving this equation is understanding and applying the Zero Product Property. The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In mathematical terms, if AB = 0, then A = 0 or B = 0 (or both). This property is a fundamental concept in algebra and is crucial for solving many types of equations, particularly those in factored form.

Applying the Zero Product Property

In the given equation, (x+3)(x+17)=0, we have two factors: (x+3) and (x+17). According to the Zero Product Property, for the product of these two factors to equal zero, at least one of them must be zero. This gives us two separate equations to solve:

x + 3 = 0
x + 17 = 0

Each of these equations is a simple linear equation that can be easily solved by isolating x on one side of the equation. This involves performing basic algebraic operations, such as addition or subtraction, to both sides of the equation to maintain balance and equality.

Solving the First Equation: x + 3 = 0

To solve the equation x + 3 = 0, we need to isolate x. This can be achieved by subtracting 3 from both sides of the equation. Subtracting the same number from both sides ensures that the equation remains balanced. The steps are as follows:

x + 3 - 3 = 0 - 3

This simplifies to:

x = -3

So, the first solution to the equation is x = -3. This means that when x is equal to -3, the factor (x+3) becomes zero, and thus the entire product (x+3)(x+17) becomes zero, satisfying the original equation.

Solving the Second Equation: x + 17 = 0

Similarly, to solve the equation x + 17 = 0, we need to isolate x. This can be done by subtracting 17 from both sides of the equation. The process is similar to the previous equation, ensuring that we maintain the equality:

x + 17 - 17 = 0 - 17

This simplifies to:

x = -17

Therefore, the second solution to the equation is x = -17. This indicates that when x is equal to -17, the factor (x+17) becomes zero, and consequently, the entire product (x+3)(x+17) is zero, satisfying the original equation.

The Solutions

We have found two values of x that satisfy the equation (x+3)(x+17)=0: x = -3 and x = -17. These are the roots or solutions of the quadratic equation. In many cases, it's helpful to express these solutions as a set: {-3, -17}. This set represents the complete solution to the equation, providing all values of x that make the equation true.

Verification of the Solutions

To ensure the accuracy of our solutions, we can verify them by substituting each value of x back into the original equation. This process confirms that the solutions indeed satisfy the equation and that no errors were made during the solving process.

Verification for x = -3

Substituting x = -3 into the original equation (x+3)(x+17)=0, we get:

(-3 + 3)(-3 + 17) = 0

(0)(14) = 0

0 = 0

This confirms that x = -3 is a valid solution because the equation holds true.

Verification for x = -17

Substituting x = -17 into the original equation (x+3)(x+17)=0, we get:

(-17 + 3)(-17 + 17) = 0

(-14)(0) = 0

0 = 0

This verifies that x = -17 is also a correct solution, as the equation is satisfied.

Alternative Methods

While using the Zero Product Property is the most efficient method for solving equations in factored form, it's worth noting that there are other approaches to solving quadratic equations. These include:

Expanding and Using the Quadratic Formula: The equation (x+3)(x+17)=0 could be expanded to the standard quadratic form ax^2 + bx + c = 0. Once in this form, the quadratic formula can be applied to find the solutions. However, this method is generally more time-consuming than using the Zero Product Property when the equation is already factored.
Completing the Square: Completing the square is another method for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial. While this method is versatile, it is often more complex than using the Zero Product Property for factored equations.

In this case, the factored form of the equation makes the Zero Product Property the most straightforward and efficient method. Recognizing the structure of the equation is crucial in choosing the best solution technique.

Real-World Applications of Quadratic Equations

Understanding how to solve quadratic equations is not just an academic exercise. Quadratic equations have numerous applications in real-world scenarios across various fields. Some examples include:

Physics: Modeling projectile motion, where the height of an object thrown into the air can be described by a quadratic equation.
Engineering: Designing structures, such as bridges and buildings, and analyzing electrical circuits often involves solving quadratic equations.
Economics: Modeling supply and demand curves, as well as cost, revenue, and profit functions.
Computer Science: Optimization problems, algorithm design, and computer graphics often utilize quadratic equations.

Having a strong grasp of how to solve quadratic equations is, therefore, a valuable skill in many professional fields.

Conclusion

In conclusion, the equation (x+3)(x+17)=0 is a quadratic equation that can be easily solved using the Zero Product Property. By setting each factor to zero and solving the resulting linear equations, we find the solutions x = -3 and x = -17. These solutions are the roots of the equation, and they can be verified by substituting them back into the original equation. Quadratic equations are a fundamental topic in mathematics with wide-ranging applications, making the ability to solve them an essential skill for students and professionals alike.

Step-by-Step Solution: Finding x in (x+3)(x+17)=0

Solving for x in the equation (x+3)(x+17)=0 is a classic problem in algebra, illustrating the application of the Zero Product Property. This property is a cornerstone in solving factored polynomials, particularly quadratic equations. The equation presented here is already factored, making the solution process more direct compared to when the equation is in its standard form (ax^2 + bx + c = 0). This detailed guide will walk you through each step, ensuring a clear understanding of the method and the underlying principles.

Understanding the Equation

The equation (x+3)(x+17)=0 is a quadratic equation in factored form. This means that the equation is expressed as the product of two binomials, (x+3) and (x+17), which equals zero. Recognizing this factored form is crucial because it allows us to apply the Zero Product Property directly. If the equation were in the standard form, we might need to factor it first, which can be a more complex process.

The Zero Product Property

The Zero Product Property is the key to solving this equation. It states that if the product of two or more factors is zero, then at least one of the factors must be zero. Mathematically, this can be expressed as: if AB = 0, then A = 0 or B = 0 (or both). This property is a direct consequence of the properties of real numbers and is a fundamental tool in algebra.

Applying the Zero Product Property to the Equation

In the equation (x+3)(x+17)=0, we have two factors: (x+3) and (x+17). According to the Zero Product Property, for this product to be zero, at least one of these factors must be zero. This gives us two separate equations to solve:

x + 3 = 0
x + 17 = 0

Each of these equations is a linear equation, which is much simpler to solve than the original quadratic equation. By solving each linear equation separately, we can find the values of x that satisfy the original equation.

Solving the First Equation: x + 3 = 0

To solve the equation x + 3 = 0, we need to isolate x on one side of the equation. This means we want to get x by itself. To do this, we can subtract 3 from both sides of the equation. Subtracting the same number from both sides maintains the balance of the equation, ensuring that the equality remains true. The steps are as follows:

x + 3 - 3 = 0 - 3

This simplifies to:

x = -3

So, the first solution to the equation is x = -3. This means that when x is equal to -3, the factor (x+3) becomes zero, and consequently, the entire product (x+3)(x+17) is zero, satisfying the original equation.

Solving the Second Equation: x + 17 = 0

Similarly, to solve the equation x + 17 = 0, we need to isolate x. This can be achieved by subtracting 17 from both sides of the equation. The process is analogous to the previous equation, and it ensures that we maintain the balance of the equation:

x + 17 - 17 = 0 - 17

This simplifies to:

x = -17

Therefore, the second solution to the equation is x = -17. This indicates that when x is equal to -17, the factor (x+17) becomes zero, and thus the entire product (x+3)(x+17) is zero, satisfying the original equation.

The Solutions Set

We have found two values of x that satisfy the equation (x+3)(x+17)=0: x = -3 and x = -17. These are the roots or solutions of the quadratic equation. It is common practice to express these solutions as a set: {-3, -17}. This set provides a clear and concise way to represent all the values of x that make the equation true.

Verification of the Solutions

To ensure the accuracy of our solutions, it is essential to verify them. This involves substituting each value of x back into the original equation to confirm that the equation holds true. This process helps to catch any potential errors that may have occurred during the solving process.

Verification for x = -3

Substituting x = -3 into the original equation (x+3)(x+17)=0, we get:

(-3 + 3)(-3 + 17) = 0

(0)(14) = 0

0 = 0

This confirms that x = -3 is a valid solution because the equation is satisfied.

Verification for x = -17

Substituting x = -17 into the original equation (x+3)(x+17)=0, we get:

(-17 + 3)(-17 + 17) = 0

(-14)(0) = 0

0 = 0

This verifies that x = -17 is also a correct solution, as the equation is satisfied.

Alternative Solution Methods

While the Zero Product Property provides the most efficient solution for equations in factored form, it is worth noting that there are other methods to solve quadratic equations. These include:

Expanding and Using the Quadratic Formula: The equation (x+3)(x+17)=0 could be expanded into the standard quadratic form ax^2 + bx + c = 0. Once in this form, the quadratic formula can be applied. However, this method is generally more complex and time-consuming for equations that are already factored.
Completing the Square: Completing the square is another method for solving quadratic equations. It involves manipulating the equation to form a perfect square trinomial. While this method is versatile, it is often less efficient than using the Zero Product Property for factored equations.

Real-World Applications of Quadratic Equations

Solving quadratic equations is not just an abstract mathematical exercise; it has numerous practical applications in various fields. Some examples include:

Physics: Modeling projectile motion, where the height of a projectile over time can be described by a quadratic equation.
Engineering: Designing structures, such as bridges and buildings, and analyzing electrical circuits often involve solving quadratic equations.
Economics: Modeling supply and demand curves, as well as cost, revenue, and profit functions.
Computer Science: Optimization problems, algorithm design, and computer graphics often utilize quadratic equations.

A solid understanding of how to solve quadratic equations is, therefore, a valuable skill in many professional domains.

Conclusion

In summary, solving the equation (x+3)(x+17)=0 using the Zero Product Property provides a clear and efficient solution. The solutions are x = -3 and x = -17, which can be expressed as the set {-3, -17}. The Zero Product Property is a fundamental concept in algebra, and its application to factored equations simplifies the solving process. Quadratic equations are a crucial topic in mathematics with wide-ranging applications, making the ability to solve them an essential skill for students and professionals alike.

x = -3, -17