Patio Expansion And The Zero Product Property Solving For X

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In the realm of mathematics, the application of algebraic principles to real-world scenarios often presents intriguing challenges. Consider the scenario of Ginger, who is grappling with the task of expanding her rectangular patio. The original dimensions of the patio are 9 feet by 6 feet, and Ginger intends to increase both the length and width by an equal amount. This expansion will result in a larger patio with an area of 88 square feet. To solve this problem, Ginger employs the zero-product property, a fundamental concept in algebra. Let's delve into the intricacies of this problem, unraveling the steps involved in finding the value of 'x' and understanding the significance of the zero-product property.

Setting the Stage The Patio Expansion Problem

The core of Ginger's problem lies in determining the value of 'x,' which represents the amount by which both the length and width of the patio are increased. Initially, the patio measures 9 feet in length and 6 feet in width. With the addition of 'x' to both dimensions, the new length becomes (9 + x) feet, and the new width becomes (6 + x) feet. The area of the expanded patio, which is the product of its length and width, is given as 88 square feet. This information leads us to the equation (6 + x)(9 + x) = 88, which Ginger intends to solve using the zero-product property.

Understanding the Zero-Product Property A Cornerstone of Algebra

The zero-product property is a powerful tool in algebra that states that if the product of two or more factors is zero, then at least one of the factors must be zero. In simpler terms, if we have an equation of the form A * B = 0, then either A = 0 or B = 0 (or both). This property is particularly useful in solving quadratic equations, which are equations of the form ax^2 + bx + c = 0. To apply the zero-product property, we must first transform the equation into a form where one side is equal to zero.

Transforming the Equation into Standard Form

Before Ginger can wield the zero-product property, she needs to manipulate the equation (6 + x)(9 + x) = 88 into a suitable form. The first step involves expanding the left side of the equation, which means multiplying the two binomials (6 + x) and (9 + x). This expansion yields: 6 * 9 + 6 * x + x * 9 + x * x = 54 + 6x + 9x + x^2. Combining like terms, we get x^2 + 15x + 54. Now, the equation looks like this: x^2 + 15x + 54 = 88. To align with the zero-product property's requirements, we need to set one side of the equation to zero. This is achieved by subtracting 88 from both sides, resulting in the equation: x^2 + 15x + 54 - 88 = 0, which simplifies to x^2 + 15x - 34 = 0.

Embracing the Quadratic Equation Factoring to Unravel the Solution

Now that Ginger has successfully transformed the equation into the standard quadratic form, x^2 + 15x - 34 = 0, she can proceed with factoring. Factoring is the process of breaking down a quadratic expression into the product of two binomials. The goal is to find two numbers that, when multiplied, give the constant term (-34) and, when added, give the coefficient of the linear term (15). After careful consideration, Ginger identifies the numbers 17 and -2 as the perfect candidates. Their product is 17 * -2 = -34, and their sum is 17 + (-2) = 15. With these numbers in hand, Ginger can rewrite the quadratic equation as: (x + 17)(x - 2) = 0.

Applying the Zero-Product Property The Moment of Truth

The stage is now set for the application of the zero-product property. Ginger recognizes that the equation (x + 17)(x - 2) = 0 represents the product of two factors, (x + 17) and (x - 2), which equals zero. According to the zero-product property, this implies that either (x + 17) = 0 or (x - 2) = 0 (or both). Ginger can now solve these two simpler equations separately.

Solving for x

Let's tackle the first equation: x + 17 = 0. To isolate 'x,' Ginger subtracts 17 from both sides, resulting in x = -17. Now, let's turn our attention to the second equation: x - 2 = 0. To isolate 'x' in this case, Ginger adds 2 to both sides, yielding x = 2. Thus, the solutions to the quadratic equation are x = -17 and x = 2.

The Significance of the Solution Interpreting the Values of x

Ginger has arrived at two potential values for 'x': -17 and 2. However, in the context of the patio expansion problem, the value of 'x' represents the amount by which the length and width of the patio are increased. It is crucial to consider whether both solutions make sense in this real-world scenario. A negative value for 'x,' such as -17, would imply that the dimensions of the patio are being decreased, which contradicts the problem's premise of expansion. Therefore, the solution x = -17 is not a feasible solution in this context.

The Practical Solution

On the other hand, the solution x = 2 represents a positive increase in both the length and width of the patio. This value aligns perfectly with the problem's description of expanding the patio. Hence, the value of 'x' that Ginger should consider is x = 2. This means that Ginger will increase both the length and width of her patio by 2 feet to achieve the desired area of 88 square feet.

Verifying the Solution A Check for Accuracy

To ensure the accuracy of her solution, Ginger can substitute x = 2 back into the original equation and verify that it holds true. The expanded dimensions of the patio would be: Length = 9 + 2 = 11 feet, Width = 6 + 2 = 8 feet. The area of the expanded patio would then be: Area = Length * Width = 11 feet * 8 feet = 88 square feet. This confirms that the solution x = 2 is indeed correct.

The Zero-Product Property in Action

In conclusion, Ginger's journey to expand her patio beautifully illustrates the power and versatility of the zero-product property. By transforming the problem into a quadratic equation, factoring the equation, and applying the zero-product property, Ginger successfully determined the value of 'x' that represents the increase in the patio's dimensions. This problem serves as a testament to the importance of algebraic principles in solving real-world problems and the elegance of mathematical solutions.

  • Zero product property
  • Quadratic equation
  • Factoring
  • Patio expansion
  • Algebraic principles