Nature of Roots

Roots are the solutions of an equation. The Nature of Roots in mathematics refers to the characteristics and properties of solutions to algebraic equations. These roots represent the values that make the equation true. Understanding the nature of roots is essential for solving equations in science and engineering to analyzing data in statistics. Depending on the equation, roots can be real or complex, and their behavior can provide insights into mathematical relationships. Our context of root in this article is for Quadratic Equations. Nature of Roots is important for Class 10 students.

In this article, we will learn about what are the roots of a quadratic equation, how to determine the nature of roots of a quadratic equation specifying different cases, and solve examples based on the nature of roots.

What are the Roots of Quadratic Equation?

In the context of quadratic equations, the term “roots” refers to the values of the variable (usually denoted as “x”) that satisfy the equation, making it true. We know that the standard representation of a Quadratic Equation is given as ax² + bx + c = 0. The roots of a quadratic equation are the values of “x” that, when substituted into the equation, make the equation true (i.e., equal to zero). There can be zero, one, or two real roots (values of “x”) depending on the discriminant (the value inside the square root) of the equation.

The roots of a Quadratic Equations is calculated using Quadratic Formula given below:

x = (-b ± √D)/2a

Where,

• b is coffecicent of x,
• D is Discriminant, and
• a is coefficient of x².

In the above formula it is the Value of Discriminant that determines the nature of roots of a quadratic equation. The details of the Nature of Roots depending upon the value of discriminant of a quadratic equation has been discussed below.

Read more about Roots of Quadratic Equation.

Nature of Roots of Quadratic Equation

This is a concept discussed in mathematics, especially when dealing with quadratic equations. The nature of the roots of a quadratic equation describes the characteristics of the “solutions” which are also known as the “roots” of that Quadratic equation. Quadratic equations are typically in the form:

Discriminant Formula

The nature of the roots for a quadratic equation given as ax² + bx + c is determined by the discriminant (D), which is calculated as:

D = b² – 4ac

Based on the value of the Discriminant (D), you can determine the nature of the roots as follows.

The value of Discriminant obtained is used to calculate the roots of a quadratic equation which is done by using quadratic formula given as

x = (-b±√D)/2a

Learn more about Discriminant Formulas for Quadratic Equations.

Different Cases of Nature of Roots

The nature of roots depends on the value of the Discriminant obtained for a given quadratic equation. Hence, the different cases of the nature of roots has been listed below:

• D > 0
• D = 0
• D < 0
• D is Perfect Square
• D is not Perfect Square

These conditions for nature of roots have been discussed extensively in the article below:

D > 0 (Positive Discriminant)

• Two distinct real roots mean the quadratic equation has two different real solutions.
  • Here the discrimination will be positive.

D = 0 (Zero Discriminant)

• One real root: In this case, the quadratic equation has only one real solution, and this solution is repeated.
  • Here the discriminant will be equal to zero.

D < 0 (Negative Discriminant)

• No real roots: The quadratic equation has no real solutions. Instead, it has two complex (conjugate) roots, which are of the form “a + bi” and “a – bi,” where “a” and “b” are real numbers, and “i” is the imaginary unit.
  • Here the discriminant will be negative.

D is a Perfect Square

• When the discriminant (D) of a quadratic equation is a perfect square (the square of a rational number), the equation has rational (real) roots.
  • Example: If D = 25, which is 5², it’s a perfect square discriminant. The equation has real roots: x = (-b ± 5) / (2a).

D is not a Perfect Square

• When D is not a perfect square, it leads to quadratic equations with either distinct irrational roots or complex conjugate roots.
  • Example: For D = 8, which is not a perfect square, the equation has two distinct irrational roots: x = (-b ± √8) / (2a)

Nature of Roots – Summary

The whole concept of Nature of Roots discussed in the article has been summarized below:

Discriminant (D)	Nature of Roots
D > 0	Two distinct real roots
D = 0	One real root (repeated)
D < 0	Two complex (conjugate) roots
D is Perfect Square	Rational & Distinct Roots
D is not a Perfect Square	Irrational & Distinct Roots

Understanding the nature of roots is essential in various fields of mathematics and science, including algebra, calculus, and physics, as it helps determine the behavior and characteristics of solutions to quadratic equations.

Nature of Roots Solved Examples

Example 1. Find the discriminant of the quadratic equation 2x²– 3x + 1 = 0.

Solution:

Given is a Quadratic equation

In the given equation, a = 2, b = -3, and c = 1.

D = (-3)² – 4(2)(1)

⇒ D = 9 – 8

⇒ D = 1

So, the discriminant is D = 1.

As the discriminant is 1 ( Which is greater than 0), The Equation will have 2 distinct real roots.

Example 2. Find the discriminant of the quadratic equation x² + 4x + 4 = 0.

Solution:

In this equation, a = 1, b = 4, and c = 4.

D = (4)² – 4(1)(4)

⇒ D = 16 – 16

⇒ D = 0

So, the discriminant is D = 0.

As the discriminant is equal to 0, the same real roots

The roots for the above Quadratic equation are 2,2

Example 3. Find the discriminant of the quadratic equation 3x² – 6x + 9 = 0.

Solution:

In this equation, a = 3, b = -6, and c = 9.

D = (-6)² – 4(3)(9)

⇒ D = 36 – 108

⇒ D = -72

So, the discriminant is D = -72.

As the discriminant is negative (<0) the equation will have the roots both roots are complex and will be conjugate pairs.

Example 4. Find the nature of roots for the Equation: x² – 4x + 4 = 0

Solution:

In this equation x² – 4x + 4 = 0

a=1 , b=-4 and c=4.

Discriminant (D) = b² – 4ac = (-4)² – 4(1)(4) = 0

Since D = 0, the roots are real and equal.

Example 5. Find the nature of the roots for the Equation: x² + 6x + 9 = 0

Solution:

In this equation x² + 6x + 9 = 0

a=1 , b=6 and c=9

Discriminant (D) = b²– 4ac = (6)² – 4(1)(9) = 36 – 36 = 0

Since D = 0, the roots are real and equal, but they are -3, a repeated root.

Roots = -3,-3.

Example 6. Find the nature of roots for the Equation: 3x² – 2x + 1 = 0

Solution:

In this equation 3x² – 2x + 1 = 0

a=3 , b=-2 and c=1

Discriminant (D) = b² – 4ac = (-2)² – 4(3)(1) = 4 – 12 = -8

Since D < 0, the roots are complex.

Nature of Roots – Practice Questions

Q1. Determine the nature of roots for the equation 2x² – 5x + 2 = 0.

Q2. Find the nature of roots for the equation 4x² + 12x + 9 = 0.

Q3. What is the nature of roots for the equation 3x² – 7x + 4 = 0?

Q4. Determine the nature of roots for the equation x² + 6x + 9 = 0.

Q5. Find the nature of roots for the equation 6x² – 11x + 4 = 0.

Nature of Roots – FAQs

What is the Nature of Roots?

The nature of roots is the nature of solutions of a quadratic equation. Based on nature of roots, they can be real roots, complex roots, equal roots, and imaginary roots.

What is the Nature of the Roots Formula?

The nature of roots is described with the discriminant of the equation. The Discriminant formula, D = b² – 4ac, determines the nature of roots in a quadratic equation. If D > 0, there are two distinct real roots; if D = 0, there’s one real root (equal roots); and if D < 0, there are no real roots, only complex roots.

How to Find the Nature of Roots?

To find the nature of roots of a quadratic equation ax²+ bx + c = 0, Calculate the discriminant (D) using the formula: D = b²– 4ac. Analyze the value of the discriminant:

• If D > 0, the equation has two distinct real roots.
• If D = 0, the equation has one real root (equal roots).
• If D < 0, the equation has no real roots, only complex roots.

What if the Discriminant is not a Perfect Square?

If Discriminat is not a perfect square then roots are irrational and distinct

Can a Quadratic Equation have more than Two Real Roots?

No, a quadratic equation can have at most two real roots. This is a fundamental property of quadratic equations.

How can I use the Nature of Roots to solve Real-World Problems?

The nature of roots can help you make informed decisions in various fields, such as physics, engineering, economics, and computer science. It aids in understanding the behavior of systems and finding solutions to problems modeled by quadratic equations.

What if the Discriminant is a Perfect Square?

If the discriminant is a perfect square then roots are rational and distinct

What if the Coefficient of ‘a’ in a Quadratic Equation is Zero?

If ‘a’ is zero, it’s not a quadratic equation but a linear equation, and it will have a single root.

9. Can you have Complex Roots with a Positive Discriminant?

No, complex roots are associated with a negative discriminant. A positive discriminant implies two distinct real sources.

10. What we have study in Nature of Roots Class 10?

in Nature of Roots Class 10 we have tom learn the conditions for the various nature of roots. The nature of roots for class 10 has been discussed below:

• D > 0: Two distinct real roots.
• D = 0: One real root (repeated).
• D < 0: Two complex (conjugate) roots.
• D is Perfect Square: Rational & Distinct Roots
• D is not a Perfect Square: Irrational & Distinct Roots