Finding Zeros Of Quadratic Function F(x) = 16x² + 32x - 9

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In this comprehensive guide, we will walk you through the process of identifying the zeros of the quadratic function f(x) = 16x² + 32x - 9. We'll explore different methods, including factoring, using the quadratic formula, and analyzing the discriminant. By the end of this article, you'll have a solid understanding of how to find the zeros of any quadratic function and confidently solve problems like this one.

Understanding Quadratic Functions and Zeros

Before diving into the solution, it's essential to grasp the basics of quadratic functions and their zeros. A quadratic function is a polynomial function of degree two, generally expressed in the form f(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.

The zeros of a quadratic function, also known as roots or x-intercepts, are the values of x for which the function f(x) equals zero. Geometrically, these are the points where the parabola intersects the x-axis. A quadratic function can have two real zeros, one real zero (a repeated root), or no real zeros (two complex roots).

Methods for Finding Zeros

Several methods can be used 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. The zeros are then found by setting each factor equal to zero and solving for x.
  2. Quadratic Formula: This formula provides a direct solution for the zeros of any quadratic function, regardless of whether it can be factored easily. The formula is: x = (-b ± √(b² - 4ac)) / 2a
  3. Completing the Square: This method involves rewriting the quadratic expression in a form that allows you to easily isolate x. It's particularly useful for deriving the quadratic formula itself.
  4. Graphing: By plotting the graph of the quadratic function, the zeros can be visually identified as the points where the parabola intersects the x-axis. This method is less precise than the algebraic methods but can provide a good estimate of the zeros.

Solving f(x) = 16x² + 32x - 9

Now, let's apply these methods to find the zeros of the given quadratic function, f(x) = 16x² + 32x - 9.

1. Using the Quadratic Formula

The quadratic formula is a reliable method for finding the zeros of any quadratic function. For f(x) = 16x² + 32x - 9, we have a = 16, b = 32, and c = -9. Plugging these values into the quadratic formula, we get:

x = (-32 ± √(32² - 4 * 16 * -9)) / (2 * 16)

Let's simplify this step by step:

x = (-32 ± √(1024 + 576)) / 32 x = (-32 ± √1600) / 32 x = (-32 ± 40) / 32

This gives us two possible solutions:

x₁ = (-32 + 40) / 32 = 8 / 32 = 1/4 = 0.25 x₂ = (-32 - 40) / 32 = -72 / 32 = -9/4 = -2.25

Therefore, the zeros of the quadratic function are x = 0.25 and x = -2.25.

2. Attempting to Factor

Factoring is another approach, but it's not always straightforward for every quadratic. We look for two numbers that multiply to a * c* (16 * -9 = -144) and add up to b (32). While this is possible, it might not be immediately obvious. However, if we persevere, we could rewrite the equation and factor it.

3. Completing the Square

Completing the square involves rewriting the quadratic expression in a perfect square form plus a constant. While it's a valid method, it's generally more complex than the quadratic formula for this specific problem.

4. Analyzing the Discriminant

The discriminant, Δ = b² - 4ac, provides valuable information about the nature of the roots. In this case, Δ = 32² - 4 * 16 * -9 = 1600. Since the discriminant is positive, we know that there are two distinct real roots, which confirms our findings using the quadratic formula.

Conclusion

By applying the quadratic formula, we've determined that the zeros of the quadratic function f(x) = 16x² + 32x - 9 are x = 0.25 and x = -2.25. Therefore, the correct answer from the given options is C. x = -2.25. This comprehensive guide has demonstrated how to solve quadratic equations using various methods, providing you with the skills and understanding to tackle similar problems effectively.

To further enhance your understanding, let's delve deeper into each method for finding zeros of quadratic functions. We will explore the underlying principles, steps involved, and scenarios where each method is most suitable.

1. Factoring Quadratic Equations: A Comprehensive Guide

Factoring is a powerful technique for solving quadratic equations, especially when the equation can be expressed as a product of two linear factors. This method relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. Let's break down the process and explore various scenarios.

Understanding the Basics of Factoring

A quadratic equation in the standard form is given by ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Factoring involves rewriting the quadratic expression as (px + q)(rx + s), where p, q, r, and s are constants. When expanded, this product should be equivalent to the original quadratic expression.

Steps Involved in Factoring

  1. Identify the Coefficients: Determine the values of a, b, and c in the quadratic equation.
  2. Find Two Numbers: Look for two numbers that multiply to ac and add up to b. This is the crucial step, and it might require some trial and error.
  3. Rewrite the Middle Term: Rewrite the middle term (bx) using the two numbers found in the previous step. For example, if the numbers are m and n, rewrite bx as mx + nx.
  4. Factor by Grouping: Group the first two terms and the last two terms and factor out the greatest common factor (GCF) from each group.
  5. Factor out the Common Binomial: If the two groups share a common binomial factor, factor it out.
  6. Set Each Factor to Zero: Set each factor equal to zero and solve for x. These solutions are the zeros (or roots) of the quadratic equation.

Example: Factoring x² + 5x + 6 = 0

  1. Identify Coefficients: a = 1, b = 5, c = 6
  2. Find Two Numbers: We need two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3.
  3. Rewrite the Middle Term: x² + 2x + 3x + 6 = 0
  4. Factor by Grouping: x(x + 2) + 3(x + 2) = 0
  5. Factor out the Common Binomial: (x + 2)(x + 3) = 0
  6. Set Each Factor to Zero:
    • x + 2 = 0 => x = -2
    • x + 3 = 0 => x = -3

Therefore, the zeros of the quadratic equation are x = -2 and x = -3.

When Factoring Works Best

Factoring is most effective when the quadratic equation has integer coefficients and the roots are rational numbers. It's a quick and elegant method when the numbers are easily identifiable. However, not all quadratic equations can be factored easily, especially when the roots are irrational or complex. In such cases, other methods like the quadratic formula or completing the square are more suitable.

Challenges in Factoring

One of the main challenges in factoring is finding the two numbers that multiply to ac and add up to b. This can be tricky, especially when ac has many factors or when the coefficients are large. In some cases, it might be impossible to find such numbers, indicating that the quadratic equation cannot be factored using integers.

Tips for Successful Factoring

  • Practice Regularly: The more you practice factoring, the better you'll become at recognizing patterns and identifying the numbers quickly.
  • Look for Special Cases: Be aware of special cases like difference of squares (a² - b² = (a + b)(a - b)) and perfect square trinomials (a² + 2ab + b² = (a + b)²).
  • Use Trial and Error Strategically: If you can't find the numbers immediately, try listing out the factors of ac and systematically checking their sums.

2. The Quadratic Formula: A Universal Solution

The quadratic formula is a powerful tool that provides a direct solution for the zeros of any quadratic equation, regardless of whether it can be factored easily. It's a universal method that guarantees a solution, even when dealing with irrational or complex roots. Let's explore the formula and its applications in detail.

Derivation of the Quadratic Formula

The quadratic formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0. The process involves manipulating the equation to isolate x:

  1. Divide by a: Divide the entire equation by a to make the coefficient of x² equal to 1: x² + (b/a)x + (c/a) = 0
  2. Move the Constant Term: Move the constant term (c/a) to the right side of the equation: x² + (b/a)x = -c/a
  3. Complete the Square: Add the square of half the coefficient of x (which is (b/2a)²) to both sides of the equation: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²
  4. Rewrite as a Perfect Square: Rewrite the left side as a perfect square: (x + b/2a)² = -c/a + b²/4a²
  5. Simplify the Right Side: Simplify the right side by finding a common denominator: (x + b/2a)² = (b² - 4ac) / 4a²
  6. Take the Square Root: Take the square root of both sides: x + b/2a = ±√(b² - 4ac) / 2a
  7. Isolate x: Isolate x by subtracting b/2a from both sides: x = (-b ± √(b² - 4ac)) / 2a

This final expression is the quadratic formula.

The Quadratic Formula

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.

Steps for Using the Quadratic Formula

  1. Identify Coefficients: Determine the values of a, b, and c in the quadratic equation.
  2. Plug into the Formula: Substitute the values of a, b, and c into the quadratic formula.
  3. Simplify: Simplify the expression, paying close attention to the order of operations.
  4. Solve for x: Calculate the two possible values of x using the plus and minus signs.

Example: Solving 2x² - 5x + 2 = 0

  1. Identify Coefficients: a = 2, b = -5, c = 2
  2. Plug into the Formula: x = (5 ± √((-5)² - 4 * 2 * 2)) / (2 * 2)
  3. Simplify:
    • x = (5 ± √(25 - 16)) / 4
    • x = (5 ± √9) / 4
    • x = (5 ± 3) / 4
  4. Solve for x:
    • x₁ = (5 + 3) / 4 = 8 / 4 = 2
    • x₂ = (5 - 3) / 4 = 2 / 4 = 1/2

Therefore, the zeros of the quadratic equation are x = 2 and x = 1/2.

The Discriminant: Nature of the Roots

The discriminant, Δ = b² - 4ac, is the part of the quadratic formula under the square root. It provides valuable information about the nature of the roots:

  • Δ > 0: Two distinct real roots.
  • Δ = 0: One real root (a repeated root).
  • Δ < 0: Two complex roots (no real roots).

Advantages of the Quadratic Formula

  • Universality: It works for all quadratic equations.
  • Direct Solution: It provides a direct solution without the need for trial and error.
  • Nature of Roots: The discriminant reveals the nature of the roots.

When to Use the Quadratic Formula

The quadratic formula is most useful when:

  • The quadratic equation cannot be factored easily.
  • The roots are irrational or complex.
  • A quick and reliable solution is needed.

3. Completing the Square: A Step-by-Step Method

Completing the square is a technique for solving quadratic equations by rewriting them in a form that allows you to easily isolate x. This method is particularly useful for deriving the quadratic formula and for understanding the structure of quadratic expressions. Let's break down the process and explore its applications.

Understanding the Basics of Completing the Square

The goal of completing the square is to transform a quadratic expression in the form ax² + bx + c into the form a(x + h)² + k, where h and k are constants. This form makes it easy to find the vertex of the parabola and to solve the quadratic equation.

Steps Involved in Completing the Square

  1. Divide by a: If a ≠ 1, divide the entire equation by a to make the coefficient of x² equal to 1.
  2. Move the Constant Term: Move the constant term (c/a) to the right side of the equation.
  3. Add the Square of Half the Coefficient of x: Add the square of half the coefficient of x to both sides of the equation. This step is the essence of completing the square.
  4. Rewrite as a Perfect Square: Rewrite the left side as a perfect square trinomial.
  5. Solve for x: Take the square root of both sides and solve for x.

Example: Solving x² + 6x + 5 = 0 by Completing the Square

  1. Divide by a: Since a = 1, we can skip this step.
  2. Move the Constant Term: x² + 6x = -5
  3. Add the Square of Half the Coefficient of x: Half of the coefficient of x is 6/2 = 3, and its square is 3² = 9. Add 9 to both sides: x² + 6x + 9 = -5 + 9
  4. Rewrite as a Perfect Square: (x + 3)² = 4
  5. Solve for x:
    • Take the square root of both sides: x + 3 = ±2
    • Isolate x:
      • x = -3 + 2 = -1
      • x = -3 - 2 = -5

Therefore, the zeros of the quadratic equation are x = -1 and x = -5.

Applications of Completing the Square

  • Deriving the Quadratic Formula: As shown earlier, completing the square is used to derive the quadratic formula.
  • Finding the Vertex of a Parabola: The vertex form of a quadratic equation, a(x + h)² + k, directly gives the vertex of the parabola at the point (-h, k).
  • Solving Quadratic Equations: Completing the square provides an alternative method for solving quadratic equations, especially when factoring is difficult.

When Completing the Square is Most Useful

  • When deriving the quadratic formula.
  • When finding the vertex of a parabola.
  • When solving quadratic equations where factoring is not straightforward.

4. Graphing Quadratic Functions: A Visual Approach

Graphing quadratic functions provides a visual approach to finding the zeros (or roots) of a quadratic equation. The zeros are the points where the parabola, which is the graph of a quadratic function, intersects the x-axis. While this method may not always provide exact solutions, it offers valuable insights into the nature of the roots and the behavior of the function. Let's explore this method in detail.

Understanding the Graph of a Quadratic Function

A quadratic function in the standard form is given by f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The graph of this function is a parabola. The shape and position of the parabola are determined by the coefficients a, b, and c.

  • If a > 0, the parabola opens upwards.
  • If a < 0, the parabola opens downwards.
  • The vertex of the parabola is the point where the parabola changes direction. Its x-coordinate is given by x = -b / 2a.
  • The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
  • The y-intercept is the point where the parabola intersects the y-axis, which is the point (0, c).

Steps for Graphing a Quadratic Function

  1. Determine the Direction of Opening: Check the sign of a. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
  2. Find the Vertex: Calculate the x-coordinate of the vertex using the formula x = -b / 2a. Then, substitute this value into the function to find the y-coordinate of the vertex.
  3. Find the Axis of Symmetry: The axis of symmetry is the vertical line x = -b / 2a.
  4. Find the y-intercept: Set x = 0 in the function to find the y-intercept.
  5. Find the x-intercepts (Zeros): Set f(x) = 0 and solve for x. These are the zeros of the function, and they are the points where the parabola intersects the x-axis. You can use the quadratic formula or factoring to find the x-intercepts.
  6. Plot the Points: Plot the vertex, the y-intercept, and the x-intercepts (if any). Plot additional points if needed to get a better sense of the shape of the parabola.
  7. Draw the Parabola: Draw a smooth curve through the plotted points, ensuring that the parabola is symmetrical about the axis of symmetry.

Finding Zeros from the Graph

The zeros of the quadratic function are the x-coordinates of the points where the parabola intersects the x-axis. These points are also known as the x-intercepts.

  • If the parabola intersects the x-axis at two distinct points, the quadratic function has two real zeros.
  • If the parabola touches the x-axis at one point (the vertex), the quadratic function has one real zero (a repeated root).
  • If the parabola does not intersect the x-axis, the quadratic function has no real zeros (two complex roots).

Advantages of Graphing

  • Visual Representation: Graphing provides a visual representation of the function and its behavior.
  • Estimating Zeros: The zeros can be estimated from the graph, although the accuracy may be limited.
  • Understanding the Nature of Roots: The graph reveals whether the roots are real or complex and how many there are.

Limitations of Graphing

  • Accuracy: Graphing may not provide exact solutions, especially if the roots are irrational.
  • Time-Consuming: Plotting the graph can be time-consuming, especially if you need to plot many points.

When Graphing is Most Useful

  • When a visual representation of the function is needed.
  • When estimating the zeros is sufficient.
  • When understanding the nature of the roots is the primary goal.

Conclusion: Choosing the Right Method

In conclusion, finding the zeros of a quadratic function involves selecting the most appropriate method based on the characteristics of the equation. Factoring is efficient for simple equations with rational roots, while the quadratic formula provides a universal solution for all quadratic equations. Completing the square is valuable for deriving the quadratic formula and understanding the structure of quadratic expressions. Graphing offers a visual approach for estimating zeros and understanding the nature of roots.

By mastering these methods, you'll be well-equipped to tackle any quadratic equation and confidently find its zeros.