Find The Zero Of Quadratic Function F(x) = 16x^2 + 32x - 9

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Let's explore how to find the zeros of a quadratic function, using the example f(x) = 16x^2 + 32x - 9. Understanding this process is crucial for solving various mathematical problems and real-world applications. In this article, we'll break down the steps and explain the underlying concepts in a friendly and accessible way. So, let’s dive in and unlock the secrets of quadratic functions!

Understanding Quadratic Functions and Zeros

Before we jump into solving the equation, let's take a moment to understand what quadratic functions and their zeros are. This foundational knowledge will make the solution process much clearer.

A quadratic function is a polynomial function of the second degree, meaning the highest power of the variable (usually 'x') is 2. The general form of a quadratic function is:

f(x) = ax^2 + bx + c

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero (otherwise, it would be a linear function). The graph of a quadratic function is a parabola, which is a U-shaped curve. This curve can open upwards (if 'a' is positive) or downwards (if 'a' is negative).

Now, what are zeros? The zeros of a function are the values of 'x' for which the function equals zero, i.e., f(x) = 0. Graphically, these are the points where the parabola intersects the x-axis. A quadratic function can have two, one, or no real zeros. These zeros are also known as roots or solutions of the quadratic equation.

Finding the zeros of a quadratic function is a common problem in algebra, and there are several methods to do so. The most common methods include:

  • Factoring: This method involves breaking down the quadratic expression into two binomial factors.
  • Quadratic Formula: This formula provides a direct way to find the zeros, regardless of whether the quadratic expression can be factored easily.
  • Completing the Square: This method involves rewriting the quadratic expression in a form that allows you to easily take the square root.

In our case, we are given the quadratic function f(x) = 16x^2 + 32x - 9, and we need to find its zeros. We will use the quadratic formula, as it is a reliable method for any quadratic equation.

Applying the Quadratic Formula

The quadratic formula is a powerful tool for finding the zeros of any quadratic function. It states that for a quadratic equation in the form ax^2 + bx + c = 0, the solutions for 'x' are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

This formula might look intimidating at first, but it's quite straightforward to use once you understand the components. The '±' symbol means there are two possible solutions: one where you add the square root term and one where you subtract it. This reflects the fact that a quadratic equation can have up to two distinct real roots.

To apply the quadratic formula, we first need to identify the coefficients 'a', 'b', and 'c' from our quadratic function f(x) = 16x^2 + 32x - 9. Comparing this to the general form ax^2 + bx + c, we can see that:

  • a = 16
  • b = 32
  • c = -9

Now, we simply plug these values into the quadratic formula:

x = (-32 ± √(32^2 - 4 * 16 * (-9))) / (2 * 16)

Let's break this down step by step to make it easier to follow.

Step-by-Step Calculation

  1. Calculate the discriminant: The expression inside the square root, b^2 - 4ac, is called the discriminant. It tells us about the nature of the roots. If the discriminant is positive, there are two distinct real roots. If it's zero, there is one real root (a repeated root). If it's negative, there are no real roots (but there are complex roots). In our case:

    Discriminant = 32^2 - 4 * 16 * (-9) = 1024 + 576 = 1600

    Since the discriminant is 1600, which is positive, we know there are two distinct real roots.

  2. Take the square root of the discriminant:

    √1600 = 40

  3. Plug the values back into the formula:

    x = (-32 ± 40) / (2 * 16)

    x = (-32 ± 40) / 32

  4. Calculate the two possible solutions:

    • Solution 1 (using the + sign):

      x1 = (-32 + 40) / 32 = 8 / 32 = 1/4 = 0.25

    • Solution 2 (using the - sign):

      x2 = (-32 - 40) / 32 = -72 / 32 = -9/4 = -2.25

So, the zeros of the quadratic function f(x) = 16x^2 + 32x - 9 are x = 0.25 and x = -2.25.

Matching the Solution with the Options

Now that we have calculated the zeros, we need to match our solutions with the given options. The options were:

A. x = -5.25 B. x = -2.25 C. x = -1.25 D. x = -0.25

Comparing our calculated zeros (x = 0.25 and x = -2.25) with the options, we can see that option B, x = -2.25, matches one of our solutions.

Therefore, the correct answer is B. x = -2.25.

Why Other Options Are Incorrect

It's also helpful to understand why the other options are incorrect. This reinforces your understanding of the solution process and helps you avoid common mistakes.

  • Option A (x = -5.25): If we plug x = -5.25 into the function f(x) = 16x^2 + 32x - 9, we get a non-zero value. This means -5.25 is not a zero of the function.
  • Option C (x = -1.25): Similarly, plugging x = -1.25 into the function will not result in zero.
  • Option D (x = -0.25): Plugging x = -0.25 into the function also gives a non-zero result.

These options might be close to the actual zeros, but they don't satisfy the condition f(x) = 0, so they are incorrect.

Alternative Methods for Finding Zeros

While we used the quadratic formula in this example, it's worth mentioning other methods for finding zeros of quadratic functions. Knowing these alternatives can be beneficial, especially in specific cases where one method might be more efficient than others.

1. Factoring

Factoring involves rewriting the quadratic expression as a product of two binomials. For example, if we have the quadratic equation x^2 + 5x + 6 = 0, we can factor it as (x + 2)(x + 3) = 0. The zeros are then found by setting each factor equal to zero:

  • x + 2 = 0 => x = -2
  • x + 3 = 0 => x = -3

Factoring is a quick method when the quadratic expression can be easily factored. However, not all quadratic expressions can be factored easily using integers. In such cases, the quadratic formula or completing the square are more reliable.

2. Completing the Square

Completing the square involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved. This method is particularly useful for deriving the quadratic formula itself. The general steps for completing the square are:

  1. Divide the equation by the coefficient of x^2 (if it's not 1).
  2. Move the constant term to the right side of the equation.
  3. Add the square of half the coefficient of x to both sides of the equation.
  4. Factor the left side as a perfect square.
  5. Take the square root of both sides and solve for x.

While completing the square can be a bit more involved than the quadratic formula, it provides a deeper understanding of the structure of quadratic equations and is a valuable technique to have in your mathematical toolkit.

Real-World Applications of Quadratic Functions

Understanding quadratic functions and how to find their zeros isn't just an academic exercise. These concepts have numerous real-world applications in various fields. Here are a few examples:

1. Physics

Quadratic functions are used extensively in physics, particularly in describing projectile motion. The trajectory of a projectile (like a ball thrown in the air) can be modeled using a quadratic equation. Finding the zeros of this equation can help determine the range of the projectile or the time it takes to hit the ground.

2. Engineering

Engineers use quadratic functions in designing bridges, arches, and other structures. The parabolic shape described by a quadratic function is often used to distribute weight evenly and provide structural stability.

3. Economics

In economics, quadratic functions can be used to model cost, revenue, and profit functions. Finding the zeros of these functions can help businesses determine break-even points or maximize profits.

4. Computer Graphics

Quadratic functions are used in computer graphics to create curves and surfaces. Bézier curves, which are widely used in computer-aided design (CAD) and animation, are based on quadratic and cubic polynomials.

These are just a few examples, and the applications of quadratic functions extend to many other areas. The ability to solve quadratic equations and understand their properties is a valuable skill in various professional fields.

Conclusion

In this article, we walked through the process of finding the zeros of the quadratic function f(x) = 16x^2 + 32x - 9. We used the quadratic formula, which is a reliable method for solving any quadratic equation. We also discussed alternative methods like factoring and completing the square, and we explored some real-world applications of quadratic functions.

Understanding quadratic functions is a fundamental concept in algebra and has practical applications in various fields. By mastering the techniques for finding zeros and interpreting their meaning, you'll be well-equipped to tackle a wide range of mathematical problems and real-world scenarios. Keep practicing and exploring, and you'll become even more confident in your ability to work with quadratic functions!

For further learning and practice on quadratic functions, you can visit Khan Academy's Quadratic Equations Section.