Finding The Roots Of F(x) = X^2 + 2x + 2 A Step-by-Step Guide

by Admin 62 views

In mathematics, finding the roots of a quadratic equation is a fundamental concept. The roots, also known as zeros or solutions, are the values of x that make the equation equal to zero. For a quadratic equation in the form of f(x) = ax^2 + bx + c, where a, b, and c are constants, the roots can be found using various methods, including factoring, completing the square, or applying the quadratic formula. In this article, we will delve into the process of identifying the roots of the given quadratic equation, f(x) = x^2 + 2x + 2. This equation is a classic example that demonstrates the application of the quadratic formula and the nature of complex roots.

Understanding Quadratic Equations

Before we dive into solving the specific equation, let's briefly discuss quadratic equations in general. A quadratic equation is a polynomial equation of the second degree. Its general form is ax^2 + bx + c = 0, where a, b, and c are coefficients, and a is not equal to zero. The roots of a quadratic equation are the values of x that satisfy the equation. Graphically, these roots represent the points where the parabola (the graph of the quadratic equation) intersects the x-axis. However, not all quadratic equations have real roots; some have complex roots, which involve the imaginary unit i (where i is the square root of -1).

Methods for Finding Roots

There are several methods to find the roots of a quadratic equation:

  1. Factoring: This method involves expressing the quadratic expression as a product of two linear factors. It is the simplest method when applicable, but it is not always possible to factor a quadratic equation easily.
  2. Completing the Square: This method involves manipulating the equation to form a perfect square trinomial, which can then be easily solved. Completing the square is a versatile method that can be used for any quadratic equation.
  3. Quadratic Formula: The quadratic formula is a general formula that provides the roots of any quadratic equation. It is derived from the method of completing the square and is a reliable method for finding roots, especially when factoring is difficult.

Applying the Quadratic Formula

The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Where:

  • a is the coefficient of the x^2 term
  • b is the coefficient of the x term
  • c is the constant term

For the given equation, f(x) = x^2 + 2x + 2, we can identify the coefficients as follows:

  • a = 1
  • b = 2
  • c = 2

Now, we substitute these values into the quadratic formula:

x = (-2 ± √(2^2 - 4 * 1 * 2)) / (2 * 1)

x = (-2 ± √(4 - 8)) / 2

x = (-2 ± √(-4)) / 2

Since we have a negative value under the square root, we will encounter complex roots. Recall that √(-1) = i, so we can rewrite √(-4) as √(4 * -1) = √(4) * √(-1) = 2i.

Thus, the equation becomes:

x = (-2 ± 2i) / 2

Now, we can simplify by dividing both terms in the numerator by 2:

x = -1 ± i

This gives us two roots:

  • x = -1 + i
  • x = -1 - i

Analyzing the Roots

We have found that the roots of the quadratic equation f(x) = x^2 + 2x + 2 are x = -1 + i and x = -1 - i. These roots are complex conjugates, meaning they have the same real part (-1) and opposite imaginary parts (+i and -i). Complex roots occur when the discriminant (the part under the square root in the quadratic formula, b^2 - 4ac) is negative. In this case, the discriminant is 4 - 8 = -4, which confirms that the roots are complex.

Implications of Complex Roots

When a quadratic equation has complex roots, it means that the parabola representing the equation does not intersect the x-axis. This is because the roots are the x-intercepts, and complex roots indicate that there are no real x-intercepts. The parabola will either be entirely above or entirely below the x-axis. In our case, since the coefficient of the x^2 term is positive (a = 1), the parabola opens upwards, and because the roots are complex, it lies entirely above the x-axis.

Conclusion

In conclusion, we have successfully identified the roots of the quadratic equation f(x) = x^2 + 2x + 2 using the quadratic formula. The roots are x = -1 + i and x = -1 - i, which are complex conjugates. This exercise demonstrates the importance of the quadratic formula in solving quadratic equations, especially when dealing with complex roots. Understanding the nature of roots (real or complex) provides valuable insights into the behavior of the quadratic function and its graphical representation. The process of applying the quadratic formula, simplifying the expression, and interpreting the results is a crucial skill in algebra and calculus, enabling us to solve a wide range of problems involving quadratic equations.

Therefore, the correct answer is D. x = -1 + i and x = -1 - i

By applying the quadratic formula and simplifying the resulting expression, we've successfully identified the roots of the given quadratic equation. The presence of complex roots indicates that the parabola does not intersect the x-axis, further enriching our understanding of quadratic functions.

Delving deeper into the topic of quadratic equations, it's crucial to understand the relationship between the discriminant and the nature of the roots. The discriminant, represented by the expression b^2 - 4ac within the quadratic formula, serves as a key indicator of the types of roots a quadratic equation possesses. Specifically, the discriminant can reveal whether the roots are real and distinct, real and equal, or complex conjugates. A positive discriminant signifies two distinct real roots, meaning the parabola intersects the x-axis at two different points. A discriminant of zero indicates one real root (or two equal real roots), where the parabola touches the x-axis at a single point, representing the vertex of the parabola. Conversely, a negative discriminant, as seen in our example with f(x) = x^2 + 2x + 2, implies the presence of complex conjugate roots, signifying that the parabola does not intersect the x-axis. This understanding not only aids in predicting the nature of the roots but also enhances the graphical interpretation of quadratic equations.

Alternative Methods

While the quadratic formula provides a direct route to finding the roots, other methods such as completing the square offer valuable insights into the structure of quadratic equations. Completing the square involves transforming the quadratic equation into a form that includes a perfect square trinomial, making it easier to identify the roots. For the equation f(x) = x^2 + 2x + 2, completing the square involves adding and subtracting a constant to create a perfect square. We start by taking half of the coefficient of the x term (which is 2), squaring it (resulting in 1), and then adding and subtracting this value within the equation: f(x) = (x^2 + 2x + 1) + 2 - 1. The expression within the parentheses is now a perfect square trinomial, which can be written as (x + 1)^2. Thus, the equation becomes f(x) = (x + 1)^2 + 1. Setting this equal to zero, we get (x + 1)^2 = -1. Taking the square root of both sides, we find x + 1 = ±√(-1), leading to x = -1 ± i, which confirms our earlier result obtained using the quadratic formula. This method not only helps in finding the roots but also reveals the vertex form of the parabola, which is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex. In this case, the vertex is (-1, 1), illustrating that the parabola opens upwards and has a minimum value at y = 1, further solidifying why the roots are complex.

The Significance of Root Types

The type of roots a quadratic equation possesses significantly influences its applications in various fields. Real roots are essential in scenarios where solutions need to be physically realizable, such as in engineering design or physics problems. For instance, calculating the trajectory of a projectile involves solving a quadratic equation, and real roots would represent the points where the projectile hits the ground. On the other hand, complex roots arise in contexts where oscillating or cyclical behavior is modeled, such as in electrical circuits or quantum mechanics. Complex numbers and their roots play a crucial role in describing phenomena that cannot be fully captured by real numbers alone. The ability to identify and interpret these roots is, therefore, a cornerstone of mathematical literacy, enabling problem-solving across a broad spectrum of disciplines. Furthermore, understanding the nature of roots helps in graphical analysis, as real roots correspond to x-intercepts, while complex roots indicate the absence of such intercepts, leading to different shapes and positions of the parabola in the Cartesian plane.

In summary, finding the roots of a quadratic equation is a fundamental skill in mathematics with far-reaching applications. Through methods such as the quadratic formula and completing the square, we can effectively determine the roots, whether they are real or complex. The nature of these roots, as dictated by the discriminant, provides crucial insights into the behavior of the quadratic function and its graphical representation. Mastering these concepts empowers us to solve a wide array of problems, from theoretical mathematics to practical applications in science and engineering. The exploration of f(x) = x^2 + 2x + 2 not only exemplifies the process of root finding but also highlights the deeper connections between algebra, geometry, and various scientific disciplines.