Factoring Quadratics Find Garden Width

In the realm of mathematics, specifically algebra, the ability to factor quadratic expressions is a fundamental skill with practical applications in various real-world scenarios. One such application lies in determining the dimensions of geometric shapes, such as gardens, given their area. This article delves into the process of factoring quadratic expressions to find the possible binomial representations of the width of a rectangular garden, given its area.

The problem at hand involves a rectangular garden with an area expressed as a quadratic expression: $(x^2 + x - 30)$ square feet. Our objective is to identify a binomial that could represent the width of the garden. To achieve this, we must first understand the relationship between the area, length, and width of a rectangle, and then apply factoring techniques to the given quadratic expression.

Understanding the Relationship Between Area, Length, and Width

The area of a rectangle is calculated by multiplying its length and width. Mathematically, this can be expressed as:

Area = Length × Width

In our case, the area of the rectangular garden is given as $(x^2 + x - 30)$ square feet. We need to find two binomials that, when multiplied together, yield this quadratic expression. These binomials will represent the possible length and width of the garden.

Factoring Quadratic Expressions: A Step-by-Step Approach

Factoring a quadratic expression involves breaking it down into its constituent binomial factors. For a quadratic expression in the form of $ax^2 + bx + c$, we seek two binomials of the form $(px + q)$ and $(rx + s)$ such that:

$(px + q)(rx + s) = ax^2 + bx + c$

To factor the quadratic expression $(x^2 + x - 30)$, we follow these steps:

1. Identify the coefficients: In this expression, $a = 1$, $b = 1$, and $c = -30$.

2. Find two numbers that multiply to c and add up to b: We need to find two numbers that multiply to -30 and add up to 1. These numbers are 6 and -5.

3. Rewrite the middle term using these numbers: We can rewrite the expression as:

   $x^2 + 6x - 5x - 30$

4. Factor by grouping: Group the first two terms and the last two terms:

   $(x^2 + 6x) + (-5x - 30)$

   Factor out the greatest common factor (GCF) from each group:

   $x(x + 6) - 5(x + 6)$

5. Factor out the common binomial factor: Notice that both terms now have a common binomial factor of $(x + 6)$. Factor this out:

   $(x + 6)(x - 5)$

Therefore, the factored form of the quadratic expression $(x^2 + x - 30)$ is $(x + 6)(x - 5)$.

Determining the Possible Width of the Garden

The factored expression $(x + 6)(x - 5)$ represents the product of the length and width of the rectangular garden. Therefore, the possible binomials that could represent the width of the garden are $(x + 6)$ and $(x - 5)$.

Now, let's examine the given answer choices:

A. $(x + 10)$ B. $(x - 10)$ C. $(x + 6)$ D. $(x - 5)$

Comparing these choices with the factors we obtained, we find that options C $(x + 6)$ and D $(x - 5)$ are the binomials that could represent the width of the garden.

Importance of Understanding the Context and Constraints

While both $(x + 6)$ and $(x - 5)$ are mathematically valid binomial factors, it's crucial to consider the context of the problem. In this case, we are dealing with the dimensions of a garden, which must be positive values. Therefore, we need to ensure that both the length and width, represented by the binomials, are positive.

If $x$ is a value such that $(x - 5)$ is negative, then this binomial cannot represent the width of the garden. However, if $x$ is greater than 5, both $(x + 6)$ and $(x - 5)$ will be positive, and either binomial could represent the width. This highlights the importance of considering the practical constraints of the problem when interpreting mathematical solutions.

Additional Examples and Applications

To further solidify the understanding of factoring quadratic expressions and its applications, let's consider a few additional examples:

Example 1:

The area of a rectangular plot of land is given by the expression $(2x^2 + 7x + 3)$ square meters. Find the possible dimensions of the plot.

1. Factor the quadratic expression:

   $(2x^2 + 7x + 3) = (2x + 1)(x + 3)$

2. Possible dimensions: The possible dimensions of the plot are $(2x + 1)$ meters and $(x + 3)$ meters.

Example 2:

The area of a rectangular frame is represented by the expression $(x^2 - 4x - 21)$ square inches. Determine the possible dimensions of the frame.

1. Factor the quadratic expression:

   $(x^2 - 4x - 21) = (x - 7)(x + 3)$

2. Possible dimensions: The possible dimensions of the frame are $(x - 7)$ inches and $(x + 3)$ inches. Again, we need to ensure that $x$ is greater than 7 for both dimensions to be positive.

These examples demonstrate the versatility of factoring quadratic expressions in solving practical problems related to geometry and spatial measurements. The ability to factor quadratic expressions empowers us to analyze and understand various real-world scenarios involving rectangular shapes and their dimensions.

Conclusion

In conclusion, factoring quadratic expressions is a valuable mathematical skill with applications in diverse areas, including geometry. By understanding the relationship between the area, length, and width of a rectangle, and by mastering factoring techniques, we can effectively determine the possible dimensions of rectangular shapes given their area. In the case of the rectangular garden with an area of $(x^2 + x - 30)$ square feet, the binomials $(x + 6)$ and $(x - 5)$ could represent the width of the garden, provided that the value of x ensures positive dimensions. This comprehensive guide has provided a step-by-step approach to factoring quadratic expressions, along with illustrative examples and a discussion of the importance of considering contextual constraints when interpreting mathematical solutions. By applying these principles, individuals can confidently tackle problems involving quadratic expressions and their real-world applications.