Location of Roots

When you solve a given equation for a particular variable, you are finding what is known as the root of the equationA root of the function f (x) is a value of x for which f (x) = 0. The graph corresponding to y = f (x) will cross the X-axis at points corresponding to the location of roots of the function.

Location of Roots Location of Roots

Create learning materials about Location of Roots with our free learning app!

  • Instand access to millions of learning materials
  • Flashcards, notes, mock-exams and more
  • Everything you need to ace your exams
Create a free account
Contents
Table of contents

    Consider the equation y = (x + 3) (x-2).

    What are the roots of the above equation?

    Solution

    The roots of the above equation are values of x for which y = 0.

    Therefore,

    y = (x + 3) (x-2) = 0

    when

    x + 3 = 0, \(\rightarrow\) x = -3

    or, x - 2 = 0 \(\rightarrow\) x = 2

    Thus, the roots of the given equation are x = 2 and x = -3.

    Location of roots theorem

    Consider the following equation:

    y = (x-2) (x + 4) (x-6)

    The following graph shows the corresponding curve.

    Location of Roots, Example graph location of roots theorem, StudySmarterGraph for the function y = (x-2) (x + 4) (x-6)

    From the above graph, we can see that the points at which the curve intersects the X-axis are at x = -4, x = 2 and x = 6. Thus, the roots of the given equation are -4, 2 and 6.

    Now look at the points on the graph marked A (corresponding to x = 1) and B (corresponding to x = 4). A lies above the X-axis, and B lies below the X-axis. Given that the graph is continuous between A and B (there is an unbroken line connecting A and B on the graph), this implies that there necessarily has to be at least one root between A and B. For the curve to travel from above the X-axis to below it, it has to cross the X-axis at some point.

    The graph being continuous in the interval between A and B is a necessary condition here. If the graph were discontinuous, there would be no compulsion for the line to intersect the X-axis. For example, you could have a function that diverges at a vertical asymptote in the given interval.

    We can generalise the above discussion to obtain the location of roots theorem:

    If the function f (x) is continuous in the interval [a, b] and f (a) and f (b) have opposite signs, then f (x) has at least one root, x, that lies between a and b, i.e. a < x < b.

    Satisfaction of the above condition means that there is at least one root between \(\x = a\) and \(\x = b\). However, it does not necessarily mean that there is only one root between \(\x = a\) and \(\x = b\). For example, consider points C and D in the above graph. C and D satisfy the condition that they are of opposite signs (the value of the function is positive at C and negative at D), but we can see from the graph that there are three roots between C and D and not just one.

    Conversely, just because two points lie on the same side of the X-axis (i.e. the value of f (x) has the same sign for two values of x) does not necessarily mean that there are no roots between them. Consider points A and C on the graph. Both are above the X-axis, i.e. the value of f (x) is positive at both points. However, we see that there are two distinct roots (at \(\x = 2\) and \(\x = 6\)) between these two points.

    Applications of the location of roots theorem

    The application of the location of roots theorem cannot be used directly to find the exact root(s) of a function. However, it can be very useful for estimating the approximate location of the roots of a function. In many methods, the location of roots theorem is applied to find an initial approximation of the roots of a function. The successive application of the theorem is used to iteratively get closer to the function's root(s). Check out our article on Iterative Methods for more details.

    In the following section, we will solve some example problems on the location of roots.

    Problems about the location of roots

    Example 1

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

    Solution 1

    f(-2) = -2³ - (-2) + 5 = -1

    f(-1) = -1³ - (-1) + 5 = 5

    Since f(-2) is negative, and f(-1) is positive, according to the Location of Roots theorem, this implies that there is at least one root of f(x) between -2 and -1.

    Example 2

    Given f (x) = x³ - 4x² + 3x + 1, show that f (x) has a root between 1.4 and 1.5

    Solution 2

    f (1.4) = 1.4³ - 4 x 1.4² + 3 x 1.4 + 1 = 0.104

    f (1.5) = 1.5³ - 4 x 1.5² + 3 x 1.5 + 1 = -0.125

    Since f(1.4) is positive, and f(1.5) is negative, according to the Location of Roots theorem, this implies that there is at least one root of f(x) between 1.4 and 1.5.

    Example 3

    For a quadratic function f(x), f(2) = 3.6, f(3) = -2.2, f(4) = -0.1, f(5) = 0.9.

    From the above information can we conclude if there is a root between

    a) 2 and 3 ?

    b) 3 and 4 ?

    c) 4 and 5 ?

    Solution 3

    a) We see that there is a change of sign between f(2) (positive) and f(3) (negative). So we can say that there is at least one root of f(x) between 2 and 3.

    b) We see that there is no change of sign between f(3) (negative) and f(4) (negative). If there has to be a root between f(3) and f(4), there would need to be at least two roots since the sign would have to change to positive and then back to negative. But we know that quadratic equations have at most two roots, and we have already found a different location for one root. This means that there is no root of f(x) between 3 and 4.

    c) We see that there is a change of sign between f(4) (negative) and f(5) (positive). So we can say that the remaining root of the quadratic, f(x) lies between 4 and 5.

    Location of Roots - Key takeaways

    • A root of the function f(x) is a value of x for which f(x) = 0.
    • The graph corresponding to y = f(x) will cross the X-axis at points corresponding to the location of roots of the function.
    • The Location of Roots theorem states that: If the function f(x) is continuous in the interval [a, b] and f (a) and f (b) have opposite signs, then f(x) has at least one root, x, that lies between a and b, i.e. a < x < b.
    • Satisfaction of the location of roots theorem means that there is at least one root between x = a and x = b. However, it does not necessarily mean that there is only one root between x = a and x = b.
    Frequently Asked Questions about Location of Roots

    How do you find the position of roots?

    If the function f(x) is continuous between a and b, and f(a) and f(b) have opposite signs, then f(x) has at least one root, x, that lies between a and b.

    What is the location of roots theorem?

    If the function f(x) is continuous in the interval [a,b] and f(a) and f(b) have opposite signs, then f(x) has at least one root, x, that lies between a and b ie. a < x < b.

    What is distinct root?

    If all roots of the equation, are different or unequal, they are called distinct roots.

    Discover learning materials with the free StudySmarter app

    Sign up for free
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Math Teachers

    • 6 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App

    Get unlimited access with a free StudySmarter account.

    • Instant access to millions of learning materials.
    • Flashcards, notes, mock-exams, AI tools and more.
    • Everything you need to ace your exams.
    Second Popup Banner