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 and for analyzing data in statistics. Depending on the equation, roots can be real or complex, and their behavior can provide insights into mathematical relationships.
What are the roots of the 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 equation are calculated using the quadratic formula given below:
x = \frac{-b \pm \sqrt{D}}{2a} Where,
- b is coefficient of x,
- D is Discriminant, and
- a is coefficient of x2.
In the above formula it is the value of the discriminant that determines the nature of the roots of a quadratic equation. The details of the nature of roots depending upon the value of the discriminant of a quadratic equation have been discussed below.
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 the discriminant obtained is used to calculate the roots of a quadratic equation, which is done by using quadratic formula given as
x = \frac{-b \pm \sqrt{D}}{2a}
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 have been listed below:
- D > 0
- D = 0
- D < 0
- D is a perfect square.
- D is not a perfect square.
These conditions for the 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)
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Nature of Roots Solved Examples
Example 1. Find the nature of the 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 2. 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 3. 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 quadratic equation has real and equal roots.
The roots for the above Quadratic equation are -2, -2.
Example 4. Find the discriminant of the quadratic equation 2x - 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 5. 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.
Example 6. 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.
Nature of Roots—Practice Questions
Q1. Determine the nature of the 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 the roots for the equation 3x2 - 7x + 4 = 0?
Q4. Determine the nature of the roots for the equation x²2 + 6x + 9 = 0.
Q5. Find the nature of roots for the equation 6x2 - 11x + 4 = 0.