The Roots Of The Function $f(x)=x^2-2x-3$ Are $x=-1$ And $x=?$
In mathematics, particularly in algebra, finding the roots of a function is a fundamental concept. The roots, also known as zeros, are the values of x for which the function f(x) equals zero. In simpler terms, they are the points where the graph of the function intersects the x-axis. Understanding how to find roots is crucial for solving various mathematical problems and has practical applications in fields like physics, engineering, and economics. This article will delve into the process of finding the roots of a quadratic function, specifically the function f(x) = x² - 2x - 3. We will explore different methods, including factoring, using the quadratic formula, and graphical approaches. By the end of this guide, you will have a solid understanding of how to determine the roots of quadratic equations and interpret their significance.
Before we dive into the specifics of finding the roots of f(x) = x² - 2x - 3, let's first understand the general form and characteristics of quadratic functions. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable x is 2. The standard form of a quadratic function is given by:
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 parabola can open upwards (if a > 0) or downwards (if a < 0). The roots of the quadratic function are the x-values where the parabola intersects the x-axis. These points are also known as the x-intercepts. A quadratic function can have two distinct real roots, one repeated real root, or no real roots (in which case the roots are complex numbers). The discriminant, given by the formula Δ = b² - 4ac, determines the nature of the roots. If Δ > 0, there are two distinct real roots; if Δ = 0, there is one repeated real root; and if Δ < 0, there are no real roots. Understanding these basic concepts is essential for effectively finding and interpreting the roots of quadratic functions.
The function we're examining is f(x) = x² - 2x - 3. One of the most straightforward methods for finding the roots of a quadratic equation is factoring. Factoring involves rewriting the quadratic expression as a product of two binomials. To factor x² - 2x - 3, we need to find two numbers that multiply to the constant term (-3) and add up to the coefficient of the linear term (-2). Let's consider the factors of -3:
- 1 and -3
- -1 and 3
Among these pairs, the pair 1 and -3 satisfies our condition because 1 * (-3) = -3 and 1 + (-3) = -2. Therefore, we can rewrite the quadratic expression as:
x² - 2x - 3 = (x + 1)(x - 3)
Now, to find the roots, we set the factored expression equal to zero:
(x + 1)(x - 3) = 0
According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for x:
- x + 1 = 0
x = -1 - x - 3 = 0
x = 3
Thus, the roots of the function f(x) = x² - 2x - 3 are x = -1 and x = 3. Factoring is a powerful technique, especially when the quadratic expression is easily factorable. It provides a clear and concise way to find the roots without resorting to more complex methods. However, not all quadratic equations are easily factorable, which is where other methods like the quadratic formula come into play.
When factoring is not straightforward or possible, the quadratic formula provides a universal method for finding the roots of any quadratic equation. The quadratic formula is derived from the standard form of a quadratic equation, ax² + bx + c = 0, and is given by:
x = (-b ± √(b² - 4ac)) / (2a)
In our example, f(x) = x² - 2x - 3, we can identify the coefficients as follows:
- a = 1 (coefficient of x²)
- b = -2 (coefficient of x)
- c = -3 (constant term)
Now, we substitute these values into the quadratic formula:
x = (-(-2) ± √((-2)² - 4(1)(-3))) / (2(1))
Simplify the expression step by step:
x = (2 ± √(4 + 12)) / 2 x = (2 ± √16) / 2 x = (2 ± 4) / 2
This gives us two possible solutions:
- x = (2 + 4) / 2 = 6 / 2 = 3
- x = (2 - 4) / 2 = -2 / 2 = -1
Thus, the roots of the function f(x) = x² - 2x - 3 are x = 3 and x = -1, which match the results we obtained through factoring. The quadratic formula is a reliable tool for finding roots, especially when dealing with quadratic equations that are difficult to factor. It is a fundamental concept in algebra and a cornerstone for solving more complex mathematical problems.
Another way to understand the roots of a function is through graphical interpretation. The roots of a function f(x) are the x-values where the graph of the function intersects the x-axis. For the quadratic function f(x) = x² - 2x - 3, the graph is a parabola. To visualize this, we can plot the function on a coordinate plane. The parabola opens upwards because the coefficient of x² (which is 1) is positive.
The roots of f(x) = x² - 2x - 3 are the points where the parabola crosses the x-axis. We have already determined that the roots are x = -1 and x = 3. This means the parabola intersects the x-axis at the points (-1, 0) and (3, 0). By plotting these points and sketching the parabola, we can visually confirm that these are indeed the roots of the function. The graphical representation provides an intuitive understanding of the roots as the x-intercepts of the parabola. It also helps to visualize the overall behavior of the quadratic function, including its vertex (the lowest point of the parabola) and axis of symmetry (the vertical line that passes through the vertex). Graphical interpretation is a valuable tool for reinforcing the algebraic concepts of roots and their significance in the context of the function's behavior.
In the given problem, we are told that one root of the function f(x) = x² - 2x - 3 is x = -1, and we are asked to find the missing root. We have already explored multiple methods for finding the roots of this quadratic function, including factoring and using the quadratic formula. Through these methods, we have consistently found that the roots are x = -1 and x = 3. Since one root is given as x = -1, the missing root must be x = 3. This can be further verified by substituting x = 3 into the function:
f(3) = (3)² - 2(3) - 3 = 9 - 6 - 3 = 0
This confirms that x = 3 is indeed a root of the function. The process of finding the missing root reinforces the understanding of how roots relate to the function's equation and its graph. It also highlights the importance of being able to apply different methods to solve the same problem, ensuring accuracy and a comprehensive understanding of the concept. The missing root, therefore, is 3.
In summary, finding the roots of a quadratic function is a critical skill in algebra and mathematics. We have explored various methods to determine the roots of the function f(x) = x² - 2x - 3, including factoring, using the quadratic formula, and graphical interpretation. Factoring allows us to rewrite the quadratic expression as a product of binomials, while the quadratic formula provides a universal solution for any quadratic equation. Graphical interpretation helps us visualize the roots as the x-intercepts of the parabola. Through these methods, we consistently found that the roots of f(x) = x² - 2x - 3 are x = -1 and x = 3. The missing root, as identified in the problem, is x = 3. Understanding these concepts and techniques not only helps in solving mathematical problems but also provides a foundation for more advanced topics in algebra and calculus. Mastering the methods for finding roots empowers students to tackle a wide range of problems and develop a deeper appreciation for the elegance and power of mathematics.