Finding Zeros Of Quadratic Function F(x) = X + 5 - 2x²

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In mathematics, finding the zeros of a function is a fundamental task. The zeros, also known as roots or x-intercepts, are the values of x for which the function f(x) equals zero. These points hold significant importance in understanding the behavior and characteristics of the function. In this article, we will delve into the process of determining the zeros of the quadratic function f(x) = x + 5 - 2x². We will explore the underlying concepts, apply relevant techniques, and arrive at the solutions while providing a comprehensive explanation that ensures clarity and understanding.

Identifying the Quadratic Function

At the outset, it is crucial to recognize that the given function f(x) = x + 5 - 2x² is a quadratic function. A quadratic function is a polynomial function of degree two, which can be expressed in the general form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. In our case, we can rewrite the function as f(x) = -2x² + x + 5, where a = -2, b = 1, and c = 5. The negative coefficient of the x² term indicates that the parabola opens downward.

Why Finding Zeros Matters

Finding the zeros of a quadratic function is not just an academic exercise; it has practical applications in various fields. The zeros represent the points where the parabola intersects the x-axis, which can be crucial in solving real-world problems. For instance, in physics, the zeros of a projectile's trajectory function can determine the range of the projectile. In engineering, they can help in designing stable structures. In economics, they can represent break-even points in cost-benefit analysis. Therefore, mastering the techniques to find zeros is essential for students and professionals alike.

Methods for Finding Zeros

There are several methods available for finding the zeros of a quadratic function, each with its own strengths and applicability. The most common methods include:

  1. Factoring: This method involves expressing the quadratic expression as a product of two linear factors. If the quadratic expression can be factored easily, this method is often the quickest and most straightforward.
  2. Completing the Square: This method involves manipulating the quadratic equation to form a perfect square trinomial, which can then be easily solved. Completing the square is a powerful technique that can be used to solve any quadratic equation, regardless of whether it can be factored.
  3. Quadratic Formula: This is a general formula that provides the solutions to any quadratic equation in the form ax² + bx + c = 0. The quadratic formula is derived by completing the square on the general form of the quadratic equation and is a reliable method for finding zeros, especially when factoring is difficult or impossible.

Choosing the Right Method

The choice of method often depends on the specific quadratic equation and the individual's preference. Factoring is efficient when the quadratic expression is easily factorable. Completing the square is a versatile method but can be more time-consuming. The quadratic formula is a foolproof method that always yields the solutions, but it may involve more calculations.

Applying the Quadratic Formula

For the given function f(x) = -2x² + x + 5, factoring might not be immediately obvious. Therefore, we will employ the quadratic formula to find the zeros. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. In our case, a = -2, b = 1, and c = 5. Substituting these values into the quadratic formula, we get:

x = (-1 ± √(1² - 4(-2)(5))) / (2(-2))

Step-by-Step Calculation

Let's break down the calculation step by step to ensure clarity.

  1. Calculate the discriminant: The discriminant is the part of the quadratic formula under the square root, b² - 4ac. In our case, the discriminant is:

    1² - 4(-2)(5) = 1 + 40 = 41

    The discriminant provides valuable information about the nature of the roots. If the discriminant is positive, the quadratic equation has two distinct real roots. If it is zero, the equation has one real root (a repeated root). If it is negative, the equation has two complex roots.

  2. Substitute the discriminant into the formula: Now we substitute the discriminant back into the quadratic formula:

    x = (-1 ± √41) / (-4)

  3. Simplify the expression: The expression can be further simplified by separating the ± sign into two separate solutions:

    x₁ = (-1 + √41) / (-4)

    x₂ = (-1 - √41) / (-4)

  4. Final Solutions: We can rewrite the solutions to make them more presentable:

    x₁ = (1 - √41) / 4

    x₂ = (1 + √41) / 4

These are the two zeros of the quadratic function f(x) = -2x² + x + 5.

Verifying the Solutions

To ensure the accuracy of our solutions, we can substitute them back into the original function and verify that f(x) = 0. While this can be a bit cumbersome with irrational numbers, it is a crucial step in confirming the correctness of our calculations.

Substituting x₁ = (1 - √41) / 4

f(x₁) = -2((1 - √41) / 4)² + ((1 - √41) / 4) + 5

Expanding and simplifying this expression, we should ideally arrive at f(x₁) = 0. Similarly, we can verify x₂.

Graphical Interpretation

Graphically, the zeros of a function represent the points where the graph of the function intersects the x-axis. For the quadratic function f(x) = -2x² + x + 5, the graph is a parabola opening downwards. The two zeros we found, x₁ = (1 - √41) / 4 and x₂ = (1 + √41) / 4, correspond to the x-coordinates of the points where the parabola intersects the x-axis. The vertex of the parabola lies midway between the zeros, and the axis of symmetry passes through the vertex.

Sketching the Graph

To sketch the graph of the parabola, we can plot the zeros and the vertex. The x-coordinate of the vertex can be found using the formula x_vertex = -b / (2a). In our case, x_vertex = -1 / (2(-2)) = 1/4. The y-coordinate of the vertex can be found by substituting x_vertex into the function: f(1/4) = -2(1/4)² + (1/4) + 5 = 41/8. Therefore, the vertex is at the point (1/4, 41/8). With the zeros and the vertex, we can sketch the parabola, which opens downwards and intersects the x-axis at x₁ and x₂.

Conclusion

In conclusion, we have successfully determined the zeros of the quadratic function f(x) = x + 5 - 2x² using the quadratic formula. The zeros are x₁ = (1 - √41) / 4 and x₂ = (1 + √41) / 4. We have also discussed the significance of finding zeros, the different methods available, and the graphical interpretation of the solutions. Understanding how to find the zeros of a function is a fundamental skill in mathematics, with applications in various fields. By mastering these techniques, students and professionals can solve real-world problems and gain a deeper understanding of the behavior of functions.

This comprehensive explanation provides a clear and detailed understanding of the process of finding the zeros of a quadratic function, ensuring that readers can grasp the concepts and apply them effectively.