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

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In the realm of mathematics, finding the zeros of a function is a fundamental task with significant implications across various fields. 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 crucial information about the function's behavior, its graph, and its relationship to the x-axis. In this article, we delve into the process of determining the zeros of the quadratic function f(x) = x + 5 - 2x², exploring the algebraic techniques involved and interpreting the significance of the solutions.

Understanding Quadratic Functions and Their Zeros

Before embarking on the quest to find the zeros of our specific function, it's essential to grasp the general nature of quadratic functions and their characteristic properties. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable x is 2. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards (if a > 0) or downwards (if a < 0).

The zeros of a quadratic function correspond to the points where the parabola intersects the x-axis. A quadratic function can have at most two real zeros, which can be distinct, repeated, or non-real (complex) numbers. The number and nature of the zeros are determined by the discriminant, a quantity derived from the coefficients of the quadratic function.

The Quadratic Formula: A Powerful Tool for Finding Zeros

To find the zeros of a quadratic function, we often employ the quadratic formula, a versatile tool that provides the solutions directly from the coefficients of the quadratic equation. The quadratic formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0. It states that the zeros of the quadratic function are given by:

x = (-b ± √(b² - 4ac)) / 2a

Where:

  • a is the coefficient of the x² term.
  • b is the coefficient of the x term.
  • c is the constant term.

The expression b² - 4ac under the square root is the discriminant, denoted by Δ. The discriminant plays a crucial role in determining the nature of the zeros:

  • If Δ > 0, the quadratic function has two distinct real zeros.
  • If Δ = 0, the quadratic function has one repeated real zero.
  • If Δ < 0, the quadratic function has two complex zeros (non-real).

Applying the Quadratic Formula to Our Function

Now, let's apply the quadratic formula to our specific function, f(x) = x + 5 - 2x². To use the formula, we first need to rewrite the function in the standard quadratic form, f(x) = ax² + bx + c:

f(x) = -2x² + x + 5

Now we can identify the coefficients:

  • a = -2
  • b = 1
  • c = 5

Plugging these values into the quadratic formula, we get:

x = (-1 ± √(1² - 4(-2)(5))) / (2(-2))

Simplifying the expression:

x = (-1 ± √(1 + 40)) / (-4)

x = (-1 ± √41) / (-4)

Therefore, the zeros of the function f(x) = x + 5 - 2x² are:

x = (-1 + √41) / (-4) and x = (-1 - √41) / (-4)

This matches option A.

Interpreting the Zeros and Their Significance

The zeros we found, x = (-1 + √41) / (-4) and x = (-1 - √41) / (-4), represent the x-coordinates of the points where the parabola defined by f(x) = -2x² + x + 5 intersects the x-axis. Since the discriminant (41) is positive, we have two distinct real zeros, indicating that the parabola crosses the x-axis at two different points.

These zeros provide valuable information about the function's behavior. For instance, they help us determine the intervals where the function is positive (above the x-axis) and negative (below the x-axis). They also play a role in finding the vertex of the parabola, which represents the maximum or minimum point of the function.

Alternative Methods for Finding Zeros

While the quadratic formula is a powerful and reliable method, there are alternative approaches for finding the zeros of quadratic functions, especially in certain cases:

  1. Factoring: If the quadratic expression can be factored into two linear factors, the zeros can be found by setting each factor equal to zero and solving for x. However, factoring is not always straightforward and may not be applicable to all quadratic equations.
  2. Completing the Square: This method involves rewriting the quadratic equation in a form that allows us to take the square root of both sides. Completing the square is also the method used to derive the quadratic formula itself.
  3. Graphing: The zeros of a quadratic function can be approximated by graphing the function and visually identifying the points where the parabola intersects the x-axis. This method is particularly useful for visualizing the zeros and understanding their relationship to the graph of the function.

Conclusion: Mastering the Art of Finding Zeros

Finding the zeros of a function is a fundamental skill in mathematics, providing insights into the function's behavior and its relationship to the x-axis. For quadratic functions, the quadratic formula offers a powerful and versatile tool for determining the zeros, regardless of the complexity of the coefficients. By understanding the quadratic formula, the discriminant, and the alternative methods available, we can confidently tackle the task of finding zeros and interpreting their significance in various mathematical contexts. In the case of f(x) = x + 5 - 2x², the zeros x = (-1 + √41) / (-4) and x = (-1 - √41) / (-4) provide valuable information about the function's graph and its behavior around the x-axis.

What are the solutions for x when the function f(x) = x + 5 - 2x² equals zero?

Finding Zeros of Quadratic Function f(x) = x + 5 - 2x²