The Significance of Finding the Roots of a Quadratic Equation

When solving an equation for a specific variable, the resulting value is known as the root of the equation. In other words, at the root, the function f (x) equals 0. The points where the graph of y = f (x) intersects the X-axis are known as the roots of the function.

Let's consider the equation y = (x + 3)(x-2). What are the roots for this equation? The roots will be the values of x where y equals 0. Therefore, when x + 3 = 0, x equals -3, and when x-2 = 0, x equals 2. Thus, the roots for this equation are x = 2 and x = -3.

Theorem on the Location of Roots

Let's take a look at the equation y = (x-2)(x+4)(x-6) and its corresponding graph. From the graph, we can see that the points where the curve intersects the X-axis are x = -4, x = 2, and x = 6. Therefore, the roots for this equation are -4, 2, and 6. Additionally, the points A (for x = 1) and B (for x = 4) on the graph show that there must be at least one root between them. This is because the graph is continuous between A and B, and for the curve to cross from above to below the X-axis, it must intersect at some point.

The continuity of the graph in the interval between A and B is a crucial factor. If the graph were discontinuous, the function may have a vertical asymptote in that interval, making it impossible to intersect with the X-axis.

Therefore, we have the following theorem on the location of roots: If the function f(x) is continuous in the interval [a, b] and f(a) and f(b) have opposite signs, then there is at least one root, x, that lies between a and b, i.e. a < x < b.

However, this theorem does not guarantee that there is only one root in an interval. For example, points C and D on the graph have opposite signs, but there are actually three roots between them, not just one. Similarly, just because two points lie on the same side of the X-axis, it doesn't mean that there are no roots between them. For instance, points A and C on the graph are both above the X-axis, but there are two distinct roots between them (at 1 and 3).

Applications of the Location of Roots Theorem

Although this theorem cannot determine the exact root(s) of a function, it can provide an estimate of their approximate location. In some methods, this theorem is used to find an initial approximation of the roots, which can then be refined further. To learn more about this, check out our article on Iterative Methods.

Solving Problems Using the Location of Roots Theorem

• Problem 1: Show that the function f(x) = x³ - x + 5 has at least one root between x = -2 and x = -1.

Solution: f(-2) = -1 and f(-1) = 5. As f (-2) is negative and f (-1) is positive, the Location of Roots theorem tells us that there is at least one root of f (x) between -2 and -1.

• Problem 2: Given f(x) = x³ - 4x² + 3x + 1, show that there is a root between 1.4 and 1.5.

Solution: f(1.4) = 0.104 and f(1.5) = -0.125. Since f(1.4) is positive and f(1.5) is negative, according to the Location of Roots theorem, there is at least one root of f(x) between 1.4 and 1.5.

• Problem 3: For a quadratic function f(x), given that f(2) = 3.6, f(3) = -2.2, f(4) = -0.1, and f(5) = 0.9, can we determine if there is a root between:

a) 2 and 3?

b) 3 and 4?

c) 4 and 5?

Solution: We can see that f(2) is positive and f(3) is negative, indicating a root between 2 and 3. Similarly, f(3) is negative and f(4) is positive, suggesting a root between 3 and 4. Lastly, f(4) is negative and f(5) is positive, meaning there is a root between 4 and 5.

The Location of Roots theorem states that if a function changes sign from positive to negative or vice versa, there must be a root between those points on the x-axis. This can be represented as < x < b.

But why is this theorem important? It allows us to determine the general location of roots in a quadratic equation. For instance, if the function f(x) shows opposite signs at points 2 and 3, we can infer that there is at least one root between these values. However, this does not necessarily mean that there is only one root in this interval. The function may switch signs multiple times, indicating the presence of more than one root.

Using the same example, if there is no sign change between f(3) and f(4), it means that there is no root between these two points. This is because a root must have a sign change from positive to negative or vice versa, and quadratic equations can have a maximum of two roots. Since we have already identified one root at a different location, there cannot be another one between f(3) and f(4).

However, if there is a sign change between f(4) and f(5), we can conclude that there is a root between these points, as the function changes from negative to positive. This means that the remaining root of the quadratic equation is located between 4 and 5 on the x-axis.

In summary, understanding the location of roots is crucial in solving and graphing quadratic equations. The Location of Roots theorem helps us determine the general location of roots between two given points. But it is important to note that there can be multiple roots within this interval.

How to Find the Location of Roots

To find the position of roots, the function f(x) must be continuous between two points, a and b, and f(a) and f(b) must have opposite signs. This allows us to apply the Location of Roots theorem and determine the general location of roots within that interval.

What is a Distinct Root?

Distinct roots refer to all the roots of a quadratic equation being different. This means that they are not equal or repeated. Understanding the concept of distinct roots is crucial in solving and analyzing quadratic equations.