Zeros Of The Quadratic Function F(x) = 8x² - 16x - 15
In the realm of mathematics, quadratic functions hold a prominent position, serving as fundamental building blocks for modeling a vast array of real-world phenomena. Understanding the characteristics of these functions, particularly their zeros, is crucial for unlocking their potential and applying them effectively. This comprehensive guide delves into the intricacies of finding the zeros of the quadratic function f(x) = 8x² - 16x - 15, providing a step-by-step approach and illuminating the underlying concepts.
The zeros of a function, also known as its roots or x-intercepts, are the points where the function's graph intersects the x-axis. At these points, the function's value is equal to zero. For quadratic functions, which are characterized by their parabolic shape, the zeros represent the x-values where the parabola crosses the horizontal axis. Finding these zeros allows us to determine the function's behavior, its minimum or maximum point, and its overall trajectory.
Why are Zeros Important?
The zeros of a quadratic function provide valuable insights into its behavior and applications. They help us:
- Determine the points where the function equals zero.
- Identify the x-intercepts of the function's graph.
- Find the axis of symmetry, which is the vertical line that divides the parabola into two symmetrical halves.
- Locate the vertex, which is the minimum or maximum point of the parabola.
- Solve real-world problems involving parabolic motion, such as the trajectory of a projectile.
With this context established, let's embark on the journey of finding the zeros of our given quadratic function, f(x) = 8x² - 16x - 15. We will explore two primary methods: the quadratic formula and completing the square, each offering its own unique approach to unraveling the function's roots.
Method 1: The Quadratic Formula - A Universal Solution
The quadratic formula stands as a cornerstone in solving quadratic equations. This powerful tool provides a direct method for finding the zeros of any quadratic function, regardless of its complexity. The formula is derived from the standard form of a quadratic equation, ax² + bx + c = 0, where a, b, and c are constants.
The quadratic formula states that the solutions for x are given by:
x = (-b ± √(b² - 4ac)) / 2a
This elegant formula elegantly captures the relationship between the coefficients of the quadratic equation and its roots. The expression under the square root, b² - 4ac, is known as the discriminant. The discriminant holds vital information about the nature of the roots:
- If b² - 4ac > 0, the equation has two distinct real roots.
- If b² - 4ac = 0, the equation has one real root (a repeated root).
- If b² - 4ac < 0, the equation has two complex roots.
Now, let's apply the quadratic formula to our function, f(x) = 8x² - 16x - 15. Here, a = 8, b = -16, and c = -15. Substituting these values into the formula, we get:
x = (-(-16) ± √((-16)² - 4 * 8 * -15)) / (2 * 8)
Simplifying this expression step-by-step, we arrive at:
x = (16 ± √(256 + 480)) / 16
x = (16 ± √736) / 16
Now, we simplify the square root of 736. The prime factorization of 736 is 2⁴ * 46, so √736 = √(16 * 46) = 4√46. Substituting this back into our equation gives:
x = (16 ± 4√46) / 16
Dividing both terms in the numerator by 4, we get:
x = (4 ± √46) / 4
Finally, separating the solutions, we obtain the two zeros of the function:
x₁ = (4 + √46) / 4 x₂ = (4 - √46) / 4
Thus, the zeros of the quadratic function f(x) = 8x² - 16x - 15, obtained using the quadratic formula, are x₁ = (4 + √46) / 4 and x₂ = (4 - √46) / 4. These values represent the precise points where the parabola intersects the x-axis, marking the function's roots with mathematical certainty.
Method 2: Completing the Square - A Journey to Vertex Form
Completing the square stands as a versatile technique, offering a pathway to rewrite a quadratic equation into vertex form. This method not only unveils the zeros of the function but also gracefully reveals the vertex, the parabola's pivotal point. Vertex form, represented as f(x) = a(x - h)² + k, provides a clear snapshot of the parabola's vertex (h, k) and its direction.
To embark on the journey of completing the square, we begin by isolating the terms containing x in our function, f(x) = 8x² - 16x - 15:
f(x) = 8(x² - 2x) - 15
Next, we focus on the expression within the parentheses, x² - 2x. Our mission is to transform it into a perfect square trinomial. To achieve this, we take half of the coefficient of our x term (-2), square it ((-1)² = 1), and add and subtract it inside the parentheses:
f(x) = 8(x² - 2x + 1 - 1) - 15
Now, we can rewrite the perfect square trinomial as a squared binomial:
f(x) = 8((x - 1)² - 1) - 15
Distributing the 8 and simplifying, we get:
f(x) = 8(x - 1)² - 8 - 15
f(x) = 8(x - 1)² - 23
We have now successfully transformed the quadratic function into vertex form, f(x) = 8(x - 1)² - 23. This form reveals that the vertex of the parabola is located at (1, -23).
To find the zeros, we set f(x) equal to zero and solve for x:
8(x - 1)² - 23 = 0
Adding 23 to both sides and dividing by 8, we get:
(x - 1)² = 23 / 8
Taking the square root of both sides, we obtain:
x - 1 = ±√(23 / 8)
Isolating x, we find the zeros:
x = 1 ± √(23 / 8)
To get rid of the fraction within the square root, we can multiply the numerator and denominator by 2:
x = 1 ± √(46 / 16)
x = 1 ± √46 / √16
x = 1 ± √46 / 4
Separating the solutions, we get:
x₁ = 1 + √46 / 4 x₂ = 1 - √46 / 4
Combining the terms into a single fraction, we have:
x₁ = (4 + √46) / 4 x₂ = (4 - √46) / 4
Therefore, the zeros of the quadratic function f(x) = 8x² - 16x - 15, as determined through the method of completing the square, are x₁ = (4 + √46) / 4 and x₂ = (4 - √46) / 4. This result harmonizes perfectly with the zeros obtained using the quadratic formula, affirming the consistency and reliability of both methods.
Comparing the Methods: A Tale of Two Approaches
Both the quadratic formula and completing the square serve as potent tools for finding the zeros of a quadratic function. However, they each possess distinct characteristics that may make one method more suitable than the other in certain situations.
The quadratic formula shines in its directness and universality. It provides a straightforward, plug-and-chug approach that works for any quadratic equation, regardless of its complexity. The formula's ability to handle any quadratic equation with ease makes it a reliable choice for those seeking a quick and efficient solution.
Completing the square, on the other hand, offers a more nuanced approach. While it may require a few more steps than the quadratic formula, it provides additional insights into the function's structure. Completing the square elegantly transforms the quadratic equation into vertex form, revealing the vertex of the parabola and its axis of symmetry. This information can be invaluable for understanding the function's behavior and sketching its graph. Furthermore, completing the square lays the foundation for understanding more advanced mathematical concepts, such as conic sections.
In essence, the choice between the quadratic formula and completing the square often boils down to personal preference and the specific context of the problem. If the primary goal is to find the zeros quickly and accurately, the quadratic formula is an excellent choice. However, if a deeper understanding of the function's structure and behavior is desired, completing the square provides a more insightful pathway.
The Zeros Unveiled: A Mathematical Triumph
In this comprehensive exploration, we have successfully unveiled the zeros of the quadratic function f(x) = 8x² - 16x - 15 using two distinct methods: the quadratic formula and completing the square. Both methods led us to the same precise solutions, x₁ = (4 + √46) / 4 and x₂ = (4 - √46) / 4, underscoring the consistency and reliability of mathematical principles.
The journey of finding the zeros has not only provided us with numerical answers but has also deepened our understanding of quadratic functions. We have witnessed how the quadratic formula serves as a universal solver, while completing the square unveils the function's vertex and structure. These insights empower us to analyze, interpret, and apply quadratic functions in a wide range of contexts.
As we conclude this exploration, let us remember that the zeros of a quadratic function are more than just numbers; they are key markers on the function's graph, revealing its intersections with the x-axis. These points provide valuable information about the function's behavior, its minimum or maximum value, and its overall trajectory. With a firm grasp of these concepts, we can confidently navigate the world of quadratic functions and harness their power to solve real-world problems.
Thus, the zeros of the quadratic function f(x) = 8x² - 16x - 15 stand as a testament to the elegance and power of mathematical methods, guiding us towards a deeper understanding of the quadratic realm.
