What Does The Value Of X Represent In The Equation (6+x)(9+x)=88 When Solving For The Increase In Dimensions Of A Rectangular Patio?

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In this article, we delve into a practical problem involving the expansion of a rectangular patio. We'll explore how to determine the amount by which the length and width must be increased to achieve a desired area, utilizing algebraic principles and the zero product property. This problem is an excellent example of how mathematical concepts can be applied to real-world scenarios, providing valuable insights for homeowners and construction professionals alike.

Problem Statement

Imagine a rectangular patio, initially measuring 9 feet in length and 6 feet in width. The homeowner decides to enlarge the patio by increasing both its length and width by the same amount. After the expansion, the patio's area becomes 88 square feet. Our goal is to determine the value of x, representing the amount by which both the length and width were increased. Ginger is using the zero product property to solve the equation (6+x)(9+x)=88(6+x)(9+x)=88. What does the value of xx represent in this context?

Setting up the Equation

To solve this problem, we'll begin by setting up an equation that represents the given information. Let's denote the amount by which both the length and width are increased as x. The new length of the patio will then be 9 + x feet, and the new width will be 6 + x feet. The area of the expanded patio can be calculated by multiplying the new length and width: (9 + x) (6 + x). We know that this new area is equal to 88 square feet. Therefore, we can write the following equation:

(9 + x) (6 + x) = 88

This equation forms the foundation for solving our problem. It represents the relationship between the original dimensions of the patio, the amount of increase (x), and the final area. To find the value of x, we need to solve this quadratic equation.

Solving the Quadratic Equation

To solve the quadratic equation (9 + x) (6 + x) = 88, we'll follow these steps:

  1. Expand the expression:

    First, we need to expand the left side of the equation by multiplying the two binomials:

    (9 + x) (6 + x) = 9 * 6 + 9 * x + 6 * x + x * x

    Simplifying this, we get:

    54 + 9x + 6x + x² = 54 + 15x + x²

  2. Rewrite the equation in standard form:

    Now, we substitute the expanded expression back into the original equation:

    54 + 15x + x² = 88

    To solve a quadratic equation, it's helpful to rewrite it in standard form, which is ax² + bx + c = 0. To do this, we subtract 88 from both sides of the equation:

    x² + 15x + 54 - 88 = 0

    Simplifying, we get:

    x² + 15x - 34 = 0

  3. Factor the quadratic expression:

    Next, we need to factor the quadratic expression x² + 15x - 34. Factoring involves finding two numbers that multiply to -34 (the constant term) and add up to 15 (the coefficient of the x term). These numbers are 17 and -2. Therefore, we can factor the expression as follows:

    (x + 17) (x - 2) = 0

  4. Apply the zero product property:

    The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, this means that either (x + 17) = 0 or (x - 2) = 0.

    Solving for x in each case:

    • If x + 17 = 0, then x = -17
    • If x - 2 = 0, then x = 2

Interpreting the Solutions

We have obtained two possible solutions for x: -17 and 2. However, in the context of this problem, x represents the amount by which the length and width of the patio are increased. Since we cannot have a negative increase in length or width, the solution x = -17 is not physically meaningful in this scenario. Therefore, we discard this solution.

The only valid solution is x = 2. This means that both the length and width of the patio were increased by 2 feet.

Verifying the Solution

To verify our solution, we can substitute x = 2 back into the original equation and check if it holds true:

(9 + x) (6 + x) = 88

(9 + 2) (6 + 2) = 88

(11) (8) = 88

88 = 88

The equation holds true, confirming that our solution x = 2 is correct.

Conclusion

In this problem, we successfully determined the amount by which the length and width of a rectangular patio were increased to achieve a desired area. By setting up a quadratic equation, solving it using factoring and the zero product property, and interpreting the solutions in the context of the problem, we found that both the length and width were increased by 2 feet. This example demonstrates the practical application of algebraic principles in solving real-world problems.

The value of x represents the increase in feet to both the length and width of the rectangular patio. In this specific context, the value of x is 2, which means both the length and the width of the patio were increased by 2 feet. This increase resulted in the patio's area expanding to 88 square feet. Understanding how to solve these types of problems is crucial for various real-world applications, from home improvement projects to more complex engineering designs.

Therefore, the value of x represents the amount, in feet, by which both the length and width of the patio were increased.

Importance of the Zero Product Property

The zero product property is a fundamental concept in algebra that plays a crucial role in solving equations, especially quadratic equations. 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 equal to zero. In simpler terms, if you have an equation like A * B = 0, then either A = 0, B = 0, or both A and B are equal to zero.

In the context of solving the patio problem, the zero product property allowed us to find the possible values of x after factoring the quadratic equation. Once we had the factored form (x + 17) (x - 2) = 0, we could apply the zero product property and set each factor equal to zero:

  • x + 17 = 0
  • x - 2 = 0

This gave us two possible solutions for x: -17 and 2. Without the zero product property, we would not be able to easily determine these solutions from the factored equation. The zero product property is essential for solving a wide range of algebraic problems, including those involving quadratic equations, polynomial equations, and other types of expressions.

Furthermore, the zero product property highlights the significance of factoring in algebra. Factoring is the process of breaking down an expression into its constituent factors, which are the building blocks of the expression. By factoring the quadratic expression in our patio problem, we transformed a complex equation into a simpler form that could be easily solved using the zero product property. Mastering factoring techniques is therefore crucial for anyone seeking to excel in algebra and related fields.

In addition to its direct applications in solving equations, the zero product property also has important implications for understanding the behavior of functions. For example, the roots or zeros of a function, which are the values of x that make the function equal to zero, can often be found by applying the zero product property to the factored form of the function. This connection between the zero product property and the roots of functions is a fundamental concept in calculus and other advanced areas of mathematics.

In conclusion, the zero product property is a cornerstone of algebra that provides a powerful tool for solving equations and understanding the properties of functions. Its importance cannot be overstated, and a thorough understanding of this property is essential for success in mathematics.

Real-World Applications of Quadratic Equations

The rectangular patio problem is just one example of how quadratic equations can be applied to real-world scenarios. Quadratic equations are mathematical expressions of the form ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. These equations arise in a wide range of fields, including physics, engineering, economics, and computer science.

In physics, quadratic equations are used to model the motion of projectiles, such as a ball thrown into the air. The height of the ball at any given time can be described by a quadratic equation, where the coefficients a, b, and c represent the effects of gravity, initial velocity, and initial height, respectively. By solving the quadratic equation, we can determine the time it takes for the ball to reach its maximum height, or the time it takes for the ball to hit the ground. This understanding is crucial in fields like sports, ballistics, and aerospace engineering.

Engineering also heavily relies on quadratic equations for designing structures, circuits, and control systems. For instance, the bending of a beam under load can be analyzed using quadratic equations, allowing engineers to determine the maximum load the beam can withstand before it fails. In electrical engineering, quadratic equations are used to calculate the impedance of a circuit, which is a measure of its resistance to alternating current. This information is essential for designing efficient and reliable electrical systems. Furthermore, in control systems, quadratic equations are used to model the behavior of feedback loops, which are used to regulate various processes, such as the temperature of a room or the speed of a motor.

In economics, quadratic equations can be used to model cost, revenue, and profit functions. For example, the cost of producing a certain number of goods may be represented by a quadratic equation, where the coefficients reflect fixed costs, variable costs, and economies of scale. Similarly, the revenue generated from selling goods may also be modeled using a quadratic equation, taking into account factors such as price elasticity of demand. By analyzing these quadratic equations, businesses can make informed decisions about pricing, production levels, and investment strategies.

Computer science utilizes quadratic equations in various algorithms and data structures. For example, in computer graphics, quadratic equations are used to represent curves and surfaces, allowing for the creation of realistic images and animations. In optimization problems, quadratic programming techniques are used to find the optimal solution to a problem that can be formulated as a quadratic equation. This has applications in areas such as machine learning, financial modeling, and logistics.

Beyond these specific examples, quadratic equations are also used in many other everyday situations. Homeowners may use them to calculate the dimensions of a garden or the amount of paint needed for a room. Financial analysts may use them to model the growth of investments or the repayment of loans. Scientists may use them to analyze experimental data or to develop theoretical models. The versatility and widespread applicability of quadratic equations make them an essential tool for problem-solving in a variety of contexts.

In conclusion, the ability to solve quadratic equations is a valuable skill that has numerous practical applications. From the simple task of expanding a patio to the complex challenges of engineering design, economic modeling, and computer science, quadratic equations provide a powerful framework for understanding and solving real-world problems.

Common Mistakes When Solving Quadratic Equations

Solving quadratic equations can be challenging, and it's easy to make mistakes if you're not careful. Understanding common errors can help you avoid them and improve your problem-solving skills. Here are some frequent mistakes students and others make when dealing with quadratic equations:

  1. Incorrectly Applying the Zero Product Property:

    One of the most common errors is misapplying the zero product property. As discussed earlier, this property states that if A * B = 0, then either A = 0 or B = 0 (or both). A frequent mistake is to apply this property when the equation is not set equal to zero. For example, if you have the equation (x + 3)(x - 2) = 5, you cannot simply set x + 3 = 5 and x - 2 = 5. Instead, you must first expand the expression, rearrange the equation to the standard form ax² + bx + c = 0, and then factor or use the quadratic formula.

  2. Factoring Errors:

    Factoring is a crucial skill for solving quadratic equations, but it can be tricky. Common factoring mistakes include:

    • Incorrectly identifying factors: For example, when factoring x² + 5x + 6, a student might incorrectly identify the factors as (x + 1)(x + 6) instead of (x + 2)(x + 3).
    • Missing a common factor: Before attempting to factor a quadratic expression, it's important to check if there's a common factor that can be factored out. For example, in the equation 2x² + 4x - 6 = 0, the common factor of 2 should be factored out first, resulting in 2(x² + 2x - 3) = 0. This simplifies the factoring process.
    • Incorrectly applying factoring patterns: Certain quadratic expressions follow specific factoring patterns, such as the difference of squares (a² - b² = (a + b) (a - b)) or perfect square trinomials (a² + 2ab + b² = (a + b)²). Forgetting or misapplying these patterns can lead to factoring errors.

  3. Errors with the Quadratic Formula:

    The quadratic formula is a powerful tool for solving quadratic equations, but it involves several steps and can be prone to errors. Common mistakes when using the quadratic formula include:

    • Incorrectly identifying coefficients: The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. Mistakes can occur when incorrectly identifying these coefficients, especially when the equation is not in standard form.
    • Sign errors: The quadratic formula involves several signs, and sign errors are a common mistake. Pay careful attention to the signs of a, b, and c when substituting them into the formula.
    • Arithmetic errors: The quadratic formula involves several arithmetic operations, including squaring, multiplication, subtraction, and division. Even a small arithmetic error can lead to an incorrect solution. It's important to double-check your calculations and use a calculator if necessary.
    • Forgetting the ± sign: The quadratic formula yields two solutions, corresponding to the plus and minus signs before the square root. Forgetting the ± sign will result in only finding one solution instead of both.

  4. Not Checking Solutions:

    After solving a quadratic equation, it's always a good practice to check your solutions by substituting them back into the original equation. This can help you identify errors in your calculations or factoring. In some cases, one or both solutions may be extraneous, meaning they don't satisfy the original equation. This can occur when dealing with equations involving square roots or fractions.

  5. Misinterpreting the Problem Context:

    In real-world problems, the solutions to a quadratic equation must be interpreted in the context of the problem. For example, in the patio problem, we obtained two solutions for x: -17 and 2. However, since x represents the amount of increase in length and width, a negative solution is not physically meaningful. Misinterpreting the problem context can lead to selecting an incorrect solution.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving quadratic equations. Practice is key to mastering this skill, so work through plenty of examples and double-check your work carefully.

In summary, solving the rectangular patio problem provides valuable insights into applying algebraic concepts to real-world scenarios. We've explored how to set up and solve a quadratic equation, interpret the solutions in context, and understand the significance of the zero product property. Additionally, we've discussed common mistakes to avoid and highlighted the broad applicability of quadratic equations across various disciplines. By mastering these concepts, individuals can enhance their problem-solving abilities and gain a deeper appreciation for the power of mathematics in everyday life.