Rewriting F(x) = X² + 8x - 20 Identifying Zeros Easily
In the realm of mathematics, quadratic functions hold a prominent position, frequently appearing in various applications across diverse fields. Understanding the behavior of these functions, particularly their zeros, is crucial for problem-solving and gaining deeper insights. The zeros of a function, also known as roots or x-intercepts, are the values of x for which the function's output, f(x), equals zero. Identifying these zeros provides valuable information about the function's graph and its relationship to the x-axis.
Our Objective:
Our primary objective is to rewrite the given quadratic function, f(x) = x² + 8x - 20, into a form that readily reveals its zeros. This involves transforming the function from its standard form to either factored form or vertex form. Factored form directly displays the zeros as the solutions to simple linear equations, while vertex form facilitates the identification of the vertex, which is essential for sketching the graph and understanding the function's symmetry. Let's embark on this journey of algebraic manipulation to unveil the hidden zeros of our quadratic function.
The Quest for Zeros: Transforming Quadratic Functions
To effectively identify the zeros of a quadratic function, we can employ two primary techniques: factoring and completing the square. Each method offers a unique pathway to rewrite the function in a more revealing form. Let's delve into each technique and explore its application to our specific function, f(x) = x² + 8x - 20.
Method 1: Factoring – Unveiling the Roots Directly
Factoring is a powerful technique that involves expressing a quadratic expression as a product of two linear expressions. When a quadratic function is in factored form, the zeros can be readily determined by setting each linear factor equal to zero and solving for x. This method directly exposes the roots of the equation, providing a clear path to understanding the function's behavior near the x-axis.
To factor the quadratic expression x² + 8x - 20, we seek two numbers that multiply to -20 (the constant term) and add up to 8 (the coefficient of the x term). After careful consideration, we identify 10 and -2 as the numbers that satisfy these conditions. Thus, we can rewrite the expression as:
x² + 8x - 20 = (x + 10)(x - 2)
By setting each factor equal to zero, we can determine the zeros:
x + 10 = 0 => x = -10 x - 2 = 0 => x = 2
Therefore, the zeros of the function f(x) = x² + 8x - 20 are x = -10 and x = 2. This factored form, f(x) = (x + 10)(x - 2), immediately reveals the zeros, making it the most convenient form for identifying these critical points.
Method 2: Completing the Square – Revealing the Vertex and Zeros
Completing the square is an algebraic technique used to transform a quadratic expression into a perfect square trinomial plus a constant term. This transformation leads to the vertex form of the quadratic function, which provides valuable information about the function's vertex (the point where the parabola changes direction) and axis of symmetry. While completing the square doesn't directly reveal the zeros as factoring does, it provides a pathway to find them by applying the square root property.
To complete the square for f(x) = x² + 8x - 20, we follow these steps:
- Focus on the x² and x terms: x² + 8x
- Take half of the coefficient of the x term (which is 8), square it (which is 16), and add and subtract it within the expression: x² + 8x + 16 - 16 - 20
- Rewrite the first three terms as a squared binomial: (x + 4)² - 16 - 20
- Simplify: (x + 4)² - 36
This gives us the vertex form of the function: f(x) = (x + 4)² - 36. The vertex of the parabola is at (-4, -36). To find the zeros, we set f(x) = 0 and solve for x:
(x + 4)² - 36 = 0 (x + 4)² = 36 x + 4 = ±√36 x + 4 = ±6 x = -4 ± 6
This yields two solutions:
x = -4 + 6 = 2 x = -4 - 6 = -10
As we found through factoring, the zeros of the function are x = -10 and x = 2. While completing the square requires an additional step to find the zeros, it provides the valuable vertex form, which is crucial for graphing and analyzing the quadratic function.
The Optimal Form: Factored Form for Zero Identification
Comparing the two methods, we can conclude that the factored form, f(x) = (x + 10)(x - 2), most easily helps identify the zeros of the function. This form directly presents the zeros as the solutions to simple linear equations. While completing the square offers valuable information about the vertex, it requires an extra step to determine the zeros.
Therefore, the answer is C. f(x) = (x + 10)(x - 2).
Delving Deeper: Why Factored Form Reigns Supreme for Zero Identification
To truly appreciate the efficiency of factored form in identifying zeros, let's delve into the underlying mathematical principles and compare it with other forms of quadratic expressions. Understanding the core concepts will solidify your grasp of this crucial aspect of quadratic functions.
The Zero Product Property: The Cornerstone of Factoring
The power of factored form stems from the zero product property, a fundamental principle in algebra. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero. Mathematically, if A * B = 0, then either A = 0 or B = 0 (or both). This property forms the bedrock of solving equations in factored form.
When a quadratic function is expressed in factored form, such as f(x) = (x + 10)(x - 2), we can directly apply the zero product property to find the zeros. By setting f(x) equal to zero, we get:
(x + 10)(x - 2) = 0
According to the zero product property, this equation holds true if either (x + 10) = 0 or (x - 2) = 0. Solving these simple linear equations immediately yields the zeros, x = -10 and x = 2. This direct link between factored form and the zeros is what makes it the most convenient form for identification.
Contrasting with Standard Form: A Less Direct Path
The standard form of a quadratic function, f(x) = ax² + bx + c, provides a general representation of the function. While it's useful for understanding the coefficients and the overall shape of the parabola, it doesn't directly reveal the zeros. To find the zeros from standard form, we typically need to employ factoring or the quadratic formula, which involve additional steps and calculations.
In our example, the standard form is f(x) = x² + 8x - 20. To find the zeros, we had to either factor it or complete the square. This highlights the indirect nature of zero identification in standard form compared to the direct approach offered by factored form.
The Vertex Form Perspective: Vertex Insights, Indirect Zeros
The vertex form, f(x) = a(x - h)² + k, provides a clear picture of the parabola's vertex (h, k) and its axis of symmetry. While this form is invaluable for graphing and understanding the function's transformations, it doesn't directly reveal the zeros. To find the zeros from vertex form, we need to set f(x) = 0 and solve for x, which involves isolating the squared term and taking the square root. This process, while straightforward, is less direct than simply reading the zeros from factored form.
Our vertex form, f(x) = (x + 4)² - 36, allowed us to find the zeros, but it required an additional step of solving a square root equation. This reinforces the notion that factored form provides the most immediate access to the zeros.
Conclusion: The Decisive Advantage of Factored Form
In the quest to identify the zeros of a quadratic function, the factored form stands out as the most efficient and direct route. The zero product property forms the foundation of this advantage, allowing us to simply set each factor to zero and solve for x. While standard form and vertex form offer valuable insights into the function's characteristics, they require additional steps to unveil the zeros.
Therefore, when the primary goal is to find the zeros, transforming the quadratic function into factored form is the most strategic approach. This technique empowers us to quickly and accurately pinpoint the roots of the equation, leading to a deeper understanding of the function's behavior and its graphical representation.
By mastering factoring and the zero product property, you'll be well-equipped to tackle a wide range of quadratic function problems and gain a profound appreciation for the elegance and power of algebraic manipulation.
