Finding The Zero Of Quadratic Function G(x) = -2x² - 5x + 1

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In mathematics, zeros of a function are the values of the input (x) that make the function's output (g(x)) equal to zero. For a quadratic function, finding these zeros is a fundamental problem with various applications in fields like physics, engineering, and economics. This article will delve into the process of finding the zero 'a' of the quadratic function g(x) = -2x² - 5x + 1. We will explore the underlying concepts, the methods used, and the practical implications of solving such problems.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, generally expressed in the form g(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens upwards if a > 0 and downwards if a < 0. The zeros of the quadratic function correspond to the x-intercepts of this parabola, the points where the curve intersects the x-axis.

In our case, the given quadratic function is g(x) = -2x² - 5x + 1. Here, a = -2, b = -5, and c = 1. Since a is negative, the parabola opens downwards, indicating that the function has a maximum value. The zeros of this function are the solutions to the equation -2x² - 5x + 1 = 0.

Methods to Find Zeros

There are several methods to find the zeros of a quadratic function, including:

  1. Factoring: This method involves expressing the quadratic expression as a product of two linear factors. If we can factor the quadratic expression, we can easily find the zeros by setting each factor equal to zero and solving for x.

  2. Completing the Square: This method involves rewriting the quadratic expression in the form a(x - h)² + k, where (h, k) is the vertex of the parabola. By setting this expression equal to zero, we can solve for x.

  3. Quadratic Formula: This is a general formula that provides the zeros of any quadratic function. It is derived from the method of completing the square and is given by:

    x = (-b ± √(b² - 4ac)) / (2a)

    This formula is particularly useful when the quadratic expression is difficult to factor.

Applying the Quadratic Formula

For the given function g(x) = -2x² - 5x + 1, we will use the quadratic formula to find the zeros. Plugging in the values a = -2, b = -5, and c = 1 into the formula, we get:

x = (-(-5) ± √((-5)² - 4(-2)(1))) / (2(-2))

x = (5 ± √(25 + 8)) / (-4)

x = (5 ± √33) / (-4)

This gives us two possible values for x, which are the zeros of the function:

x₁ = (5 + √33) / (-4) x₂ = (5 - √33) / (-4)

Thus, the zeros of the quadratic function g(x) = -2x² - 5x + 1 are approximately:

x₁ ≈ -2.686 x₂ ≈ 0.186

These values represent the points where the parabola intersects the x-axis.

Detailed Steps and Explanation

Let's break down the steps of using the quadratic formula in more detail:

  1. Identify the Coefficients:

    • First, we need to identify the coefficients a, b, and c from the quadratic equation in the form ax² + bx + c = 0. In our case, g(x) = -2x² - 5x + 1, so a = -2, b = -5, and c = 1.

  2. Write Down the Quadratic Formula:

    • The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It is given by:

      x = (-b ± √(b² - 4ac)) / (2a)

  3. Substitute the Values:

    • Next, substitute the values of a, b, and c into the quadratic formula:

      x = (-(-5) ± √((-5)² - 4(-2)(1))) / (2(-2))

  4. Simplify the Expression:

    • Simplify the expression step by step:

      • x = (5 ± √(25 + 8)) / (-4)
      • x = (5 ± √33) / (-4)

  5. Calculate the Two Possible Values of x:

    • The ± sign in the formula indicates that there are two possible solutions for x. Calculate each separately:

      • x₁ = (5 + √33) / (-4)
      • x₂ = (5 - √33) / (-4)

  6. Approximate the Values (if needed):

    • If an approximate value is required, use a calculator to find the square root of 33 and then perform the calculations:

      • x₁ ≈ (5 + 5.745) / (-4) ≈ -2.686
      • x₂ ≈ (5 - 5.745) / (-4) ≈ 0.186

The Discriminant

An important part of the quadratic formula is the discriminant, which is the expression b² - 4ac under the square root. The discriminant tells us about the nature of the roots:

  • If b² - 4ac > 0, there are two distinct real roots.
  • If b² - 4ac = 0, there is exactly one real root (a repeated root).
  • If b² - 4ac < 0, there are no real roots, but there are two complex roots.

In our case, the discriminant is (-5)² - 4(-2)(1) = 25 + 8 = 33. Since 33 > 0, we have two distinct real roots, as we found earlier.

Practical Implications and Applications

Finding the zeros of quadratic functions is not just an abstract mathematical exercise. It has numerous practical applications in various fields:

  1. Physics:

    • In physics, quadratic functions are used to model projectile motion. The zeros of the function represent the points where the projectile hits the ground. For example, if we model the height of a projectile as a function of time using a quadratic equation, the zeros of the function will give us the time when the projectile lands.

  2. Engineering:

    • Engineers use quadratic equations to design structures, circuits, and systems. For instance, in structural engineering, the shape of a parabolic arch can be modeled using a quadratic function. Finding the zeros helps determine the points where the arch meets the ground or other supports.

  3. Economics:

    • In economics, quadratic functions can model cost, revenue, and profit. The zeros of these functions can help determine break-even points, where the cost equals the revenue. For example, a business might use a quadratic cost function to determine the production level at which it starts making a profit.

  4. Computer Graphics:

    • Quadratic functions are used in computer graphics to create curves and surfaces. Bézier curves, which are widely used in graphic design and animation, are often based on quadratic or cubic functions. Finding the zeros and other critical points of these functions is essential for rendering smooth and accurate shapes.

  5. Optimization Problems:

    • Many optimization problems involve finding the maximum or minimum value of a function. For quadratic functions, the vertex of the parabola represents the maximum or minimum point. Finding the zeros can help determine the x-value at which this extremum occurs.

Conclusion

Finding the zeros of a quadratic function is a crucial skill in mathematics with wide-ranging applications. In this article, we focused on finding the zero 'a' of the quadratic function g(x) = -2x² - 5x + 1. We explored the concept of quadratic functions, the methods to find their zeros, and the practical implications of these methods. By using the quadratic formula, we found the zeros to be approximately -2.686 and 0.186. Understanding these concepts and methods is essential for students and professionals alike, as quadratic functions and their zeros appear in various contexts across different fields.

Mastering the techniques to find zeros, including factoring, completing the square, and using the quadratic formula, provides a robust foundation for tackling more complex mathematical problems. The quadratic formula, in particular, is a versatile tool that can be applied to any quadratic equation, making it an indispensable part of the mathematical toolkit.

Furthermore, understanding the nature of the roots through the discriminant allows for a deeper analysis of the quadratic function and its behavior. The discriminant helps predict the number and type of solutions, guiding the problem-solving process and enhancing the understanding of mathematical concepts.

In conclusion, the ability to find the zeros of quadratic functions is not just an academic exercise but a practical skill that empowers individuals to solve real-world problems across various disciplines. The methods and concepts discussed in this article provide a comprehensive guide to understanding and solving quadratic equations, ensuring a solid foundation for future mathematical endeavors.