Location of Roots: Quadratic Equation, Theorem, Formula, Questions

Introduction to Location of Roots

The location of roots helps determine where the solutions (roots) of a quadratic equation lie on the number line—whether they are positive, negative, real, complex, or on opposite sides of a point (like the origin or a specific number). Instead of finding the exact values of the roots, the goal is to infer their signs and relative positions using logical conditions and basic algebraic tools.

What is the Location of Roots?

Given a quadratic equation
$ax^2 + bx + c = 0$

Let the roots be $x_1$ and $x_2$.
The location of roots describes whether:

• Both roots are positive

• Both roots are negative

• Roots lie on opposite sides of zero (or any given number)

• Roots are real and distinct, real and equal, or complex

Key Formulas to Analyze Root Location

Discriminant:
$D = b^2 - 4ac$

Sum of roots:
$x_1 + x_2 = \dfrac{-b}{a}$

Product of roots:
$x_1 \cdot x_2 = \dfrac{c}{a}$

Basic Conditions for Location and Nature of Roots

• Roots are real and distinct if $D > 0$

• Roots are real and equal if $D = 0$

• Roots are complex (non-real) if $D < 0$

• Both roots are positive if $x_1 + x_2 > 0$ and $x_1 \cdot x_2 > 0$

• Both roots are negative if $x_1 + x_2 < 0$ and $x_1 \cdot x_2 > 0$

• Roots lie on opposite sides of zero if $x_1 \cdot x_2 < 0$

Standard Form of a Quadratic Equation

Quadratic equations are algebraic equations of degree 2, and understanding their form is essential before studying the location of roots. The standard form of a quadratic equation is

$ax^2 + bx + c = 0$,

where $a, b,$ and $c$ are real constants and $a \ne 0$.

To find the location of roots of a function, several well-known methods are used, such as the sign change method, graphical method to locate roots, Intermediate Value Theorem, Rolle’s Theorem, and Descartes’ Rule of Signs. These techniques help estimate where the roots lie on the number line without always solving the equation exactly.

Understanding how to find the approximate location of roots of a polynomial makes solving quadratic and higher-degree equations easier, more visual, and more intuitive—an essential skill in algebra and graph-based problem solving.

Meaning of Roots or Zeros of a Quadratic Equation

The roots (or zeros) of a quadratic equation are the values of $x$ that satisfy

$ax^2 + bx + c = 0$.

These roots represent the x-intercepts of the parabola defined by

$f(x) = ax^2 + bx + c$.

Understanding the location of roots is essential when analysing the graph of a quadratic function, as it tells us where the graph crosses or touches the x-axis.

The roots of a quadratic equation can be found using the quadratic formula:

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

To further study how to find the location of roots of a polynomial, methods such as the graphical method to locate roots, the sign change method for roots, and the Intermediate Value Theorem are widely used. These techniques help in finding the location of roots of an equation, estimating their values, and understanding whether the roots are real, repeated, or complex, without always solving the equation exactly.

Real vs Complex Roots

The discriminant $D=b^2−4ac$ determines the nature of the roots:

• If D>0, roots are real and distinct
• If D=0, roots are real and equal
• If D<0, roots are complex conjugates

This classification is crucial for analyzing where and how the roots appear on the number line or complex plane.

Conditions for Location of Roots in Quadratic Equations

To determine the location of roots of a quadratic equation, several algebraic tests based on the coefficients and discriminant are used. These methods help in understanding how to find the location of roots of a polynomial and whether the roots lie on the positive or negative side of the number line. Techniques such as the sign change method for locating roots, the Intermediate Value Theorem, and concepts related to Rolle’s Theorem and root location are commonly applied. Together, these approaches make it easier to identify the position, nature, and behavior of roots without solving the equation completely.

Discriminant and Nature of Roots $D=b^2−4ac$

Use it to decide:

• D>0: Two real and distinct roots
• D=0: Real and equal roots
• D<0: No real roots (complex)

Real roots are necessary to discuss their location on the number line.

Sign of Roots Based on Coefficients

Using formulas:

Sum of roots:
$x_1 + x_2 = \dfrac{-b}{a}$

Product of roots:
$x_1 \cdot x_2 = \dfrac{c}{a}$

Conditions:

If $x_1 + x_2 > 0$, both roots are positive.

If $x_1 + x_2 < 0$, both roots are negative.

If $x_1 \cdot x_2 < 0$, the roots lie on opposite sides of zero.

Location of Roots on the Number Line

To determine where the roots lie:

• Check sign of f(x) at a point (like f(0)=c)
• Analyze the sign of the product $x_1⋅x_2$
• Graphical approach: roots are x-intercepts of the parabola

Quick checks:

• $f(0) = c > 0$ ⇒ roots lie on opposite sides of $0$
• If $x_1 + x_2 > 0$ and $x_1 \cdot x_2 > 0$, the roots lie on the positive side.
• Whether the roots are both positive or both negative depends on the sign consistency of the function.

Importance in Algebra and Real-World Applications

The concept of location of roots is widely used in:

• Algebra: For solving inequalities and analyzing signs of expressions without solving the equation fully.
• Graphing: Understanding whether the parabola intersects the x-axis and where.
• Inequalities: For checking when $ax^2+bx+c>0$ or $<0$ based on root positions.
• Physics and Engineering: To ensure solutions (e.g., time, speed) are positive and physically meaningful.
• Economics: For identifying profit/loss break-even points based on quadratic cost/revenue functions.

Different Cases of Roots of a Quadratic Equation

The nature and location of roots of a quadratic equation $ax^2+bx+c=0$ largely depend on the discriminant $D=b^2−4ac$. Understanding this helps in finding location of roots of equation more effectively.

Different cases—real and distinct, real and equal, or complex roots—affect how to locate roots of a function both algebraically and visually. Methods like the graphical method to locate roots, Descartes' Rule of Signs, and the intermediate value theorem for roots can assist in analysing these scenarios.

Case 1: D>0 – Real and Distinct Roots

Roots are real and unequal.

If $\dfrac{c}{a} > 0$ and $\dfrac{-b}{a} > 0$, both roots are positive.

If $\dfrac{c}{a} > 0$ and $\dfrac{-b}{a} < 0$, both roots are negative.

If $\dfrac{c}{a} < 0$, the roots lie on opposite sides of the origin.

Case 2: D=0 – Real and Equal Roots

• Roots are real and repeated.

• The single root is given by $x = \dfrac{-b}{2a}$

• The root is positive if $\dfrac{-b}{2a} > 0$ and negative if $\dfrac{-b}{2a} < 0$

Case 3: D<0 – Imaginary Roots

• Roots are non-real (complex conjugate pair).
• No real root exists ⇒ no location on the number line.

Case 4: D is a Perfect Square

• Roots are real and rational
• Examples: D=4, 9, 16, etc.
• Occurs often in factorable quadratic equations

Case 5: D is Not a Perfect Square

• Roots are real and irrational.
• Cannot be simplified to a rational number.
• Roots still follow the same sign rules using sum and product.

Sum and Product of Roots of a Quadratic Equation

Let $\alpha$ and $\beta$ be the roots of the quadratic equation

$ax^2 + bx + c = 0$

Sum of Roots Formula

$\alpha + \beta = \dfrac{-b + \sqrt{D}}{2a} + \dfrac{-b - \sqrt{D}}{2a}$

$= \left(\dfrac{-b}{2a} + \dfrac{\sqrt{D}}{2a}\right) + \left(\dfrac{-b}{2a} - \dfrac{\sqrt{D}}{2a}\right)$

$= \dfrac{-2b}{2a}$

$= \dfrac{-b}{a}$

So, the sum of roots is

$\alpha + \beta = \dfrac{-b}{a}$

Product of Roots Formula

$\alpha \cdot \beta = \left(\dfrac{-b + \sqrt{D}}{2a}\right)\left(\dfrac{-b - \sqrt{D}}{2a}\right)$

$= \dfrac{(-b)^2 - (\sqrt{D})^2}{(2a)^2}$

$= \dfrac{b^2 - D}{4a^2}$

$= \dfrac{b^2 - (b^2 - 4ac)}{4a^2}$

$= \dfrac{4ac}{4a^2}$

$= \dfrac{c}{a}$

So, the product of roots is

$\alpha \cdot \beta = \dfrac{c}{a}$

Forming a Quadratic Equation from Roots

Given $\alpha + \beta$ and $\alpha \cdot \beta$, the quadratic equation is

$x^2 - (\alpha + \beta)x + (\alpha \cdot \beta) = 0$

Substituting the given values,

$x^2 - \dfrac{b}{a}x + \dfrac{c}{a} = 0$

Multiplying the entire equation by $a$,

$ax^2 - bx + c = 0$

This is the original quadratic equation reconstructed using the sum and product of roots.

Graphical Interpretation of Location of Roots

1. If both roots of f(x) are less than k then

2. If both roots of f(x) are greater than k

i) $D \ge 0$ (as the real roots may be distinct or equal)

ii) $a f(k) > 0$ (in both cases $a f(k)$ is positive; if $a < 0$, then $f(k) < 0$, and the product of two negative values is positive)

iii) $k < \dfrac{-b}{2a}$ since $\dfrac{-b}{2a}$ lies between $x_1$ and $x_2$, and $x_1, x_2$ are greater than $k$, so $\dfrac{-b}{2a}$ is also greater than $k$

Condition for number $k$

Let
$f(x) = ax^2 + bx + c$, where $a, b, c$ are real numbers and $a \ne 0$.
Let $x_1$ and $x_2$ be the real roots of the function, and let $k$ be any real number.

3. If k lies between the root $x_1$ and $x_2$

$a f(k) < 0$

As if $a < 0$, then $f(k) > 0$. So, multiplying one negative and one positive value gives a negative result.

Condition for numbers $k_1$ and $k_2$

Let
$f(x) = ax^2 + bx + c$, where $a, b, c$ are real numbers and $a \ne 0$.
Let $x_1$ and $x_2$ be the real roots of the function, and let $k_1, k_2$ be any two real numbers.

4. If exactly one root of f(x) lies in between the number $k_1,k_2$

$f(k_1) f(k_2) < 0$, as for one value of $k$ we get a positive value of $f(x)$, and for the other value of $k$ we get a negative value of $f(x)$ (here $x_1 < x_2$).

Condition on numbers $k_1, k_2$

Let
$f(x) = ax^2 + bx + c$, where $a, b, c$ are real numbers and $a \ne 0$.
Let $x_1$ and $x_2$ be the roots of the function, and let $k_1, k_2$ be any two real numbers.

5. If both roots lie between $k_1,k_2$

i) $D \ge 0$ (as the real roots may be distinct or equal)

ii) $k_1 < \dfrac{-b}{2a} < k_2$, where $a \le 3$ and $k_1 < k_2$

6. If $k_1,k_2$ lies between the roots

$a f(k_1) < 0$ and $a f(k_2) < 0$

Important Formulae and Concepts for Quadratic Equations

The following table highlights the important formulae and key concepts of quadratic equations, helping you quickly analyze the nature and location of roots without lengthy calculations.

Solved Examples Based on Location of Roots

Example 1: Number of +ve integer values of 'a' for which $x^2+x+a=0$ has roots of opposite sign is.......

Solution:

Roots are of opposite sign.

$ \dfrac{c}{a} < 0 $ and $ D = b^2 - 4ac > 0 $

$D > 0$

$\dfrac{c}{a} < 0 \Rightarrow 1 - 4a > 0$

$a - 1 < 0$

$\Rightarrow a < 1$

$\Rightarrow a < 0$

Hence, the number of positive integer values of $a$ is $0$.

Example 2: All the values of m for which both roots of the equations $x^2−2mx+m^2−1=0$ are greater than -2 but less than 4, lie in the interval:

a) −2<m<0

b) m>3

c) −1<m<3

d) 1<m<4

Solution:

Example 3: What is the value of m so that both roots of the equation $x^2+mx+1$ are less than unity?

Solution:

Since the roots are less than unity, it implies that
$a f(1) > 0$, where $a = 1$ and $f(1) = m + 2$

So,
$m + 2 > 0 \Rightarrow m > -2$ ...(i)

For two distinct real roots to exist, the discriminant must be positive:

$D > 0 \Rightarrow m^2 - 4 > 0$

So,
$m > 2$ or $m < -2$ ...(ii)

Now, combining (i) and (ii):

From (i): $m > -2$
From (ii): $m > 2$ or $m < -2$

The common values satisfying both conditions are:
$m > 2$

Hence, the final solution is:
$m > 2$

Example 4: What is the value of m so that one root of the equation $x^2−(m−3)x+m=0$ is less than 2 and the other root is greater than 2?

Solution:

If one root is less than $2$ and $f(2) < 0$

$D > 0$

$(m-3)^2 - 4m > 0$

$(m-1)(m-9) > 0$

$m \in (-\infty, 1) \cup (9, \infty)$

Now,
$f(2) < 0$

$4 - 2(m-3) + m < 0$

$10 - m < 0$

$m > 10$

On combining both conditions,

$m \in (10, \infty)$

Example 5: Find the values of a for which exactly one root of the equation $x^2+2ax+a^2−1=0$ lies between the numbers 4 and 6.

Solution:

If exactly one root of the equation
$x^2 + 2ax + a^2 - 1 = 0$
lies between $4$ and $6$, then
$f(4)\cdot f(6) < 0$

$(16 + 8a + a^2 - 1)(36 + 12a + a^2 - 1) < 0$

$(a^2 + 8a + 15)(a^2 + 12a + 35) < 0$

$(a+5)(a+3)(a+7)(a+5) < 0$

$(a+3)(a+7)(a+5)^2 < 0$

$(a+3)(a+7) < 0$

$a \in (-7,-3)$

List of Topics related to the Location of Roots

Here is a list of key topics related to the location of roots in quadratic equations and polynomials, including concepts like remainder theorem, rational and irrational inequalities, along with exponential and logarithmic equations in quadratic form.

NCERT Useful Resources

Explore essential NCERT study materials for Complex Numbers and Quadratic Equations, including detailed solutions, concise revision notes, and curated exemplar problems. These resources are tailored to help you strengthen your conceptual understanding and prepare effectively for board and competitive exams.

• NCERT Maths Notes for Class 11th Chapter 4 - Complex Numbers and Quadratic Equations

• NCERT Maths Solutions for Class 11th Chapter 4 - Complex Numbers and Quadratic Equations

• NCERT Maths Exemplar Solutions for Class 11th Chapter 4 - Complex Numbers and Quadratic Equations

Practice Questions on the location of roots

To help you strengthen your understanding of the topic, we’ve included a few practice questions on location of roots of quadratic equations. These will test your grasp of root positions, signs, and related conditions using standard formulas.

To practice location of roots-based questions, click here.

You can practice the next topics of Complex Numbers and Quadratic Equations below: