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Nature of the Roots
Before learning about the nature of the roots, firstly, we need to understand quadratic equations. A quadratic polynomial is a polynomial of degree two. A quadratic polynomial becoming equal to zero results in the formation of a quadratic equation.
In mathematics, quadratic equations are an essential aspect of Algebra. Quadratic equations have at least one term in square form. The most conventional type of quadratic equation:
ax2 + bx + c = 0
Nature of Roots of Quadratic Equation
Students frequently ask whether a quadratic equation may have several solutions. Is it possible that some equations have no true solution? The answer is yes, it is doable using the concept of the nature of roots of a quadratic equation.
It is the value of a variable for which the equation is solved. The degree of a polynomial equation is equal to the number of its roots. As a result, there are two roots to a quadratic equation. Let assume α and β represents the roots of the quadratic equation:
ax2 + bx + c = 0.
The following are the formulae for solving quadratic equations:
α = (-b-√b2-4ac)/2a β = (-b+√b2-4ac)/2a
Where a, b, and c are all rational and real numbers.
As a result, the nature of the roots α and β of the equation ax2 + bx + c = 0 is determined by the square root of b2 – 4ac because the root of a negative number can never be any real number. A quadratic equation is x2 = -1. There is no actual integer with a negative square. As a result, there are no real- number of solutions to this problem.
Type of Roots
The term ‘nature’ denotes different types of roots, such as actual, rational, irrational, or fictitious. There are three different kinds of roots of a quadratic equation.
- Complex roots
- Real and equal roots
- Real and Distinct roots
Discriminant and Nature of Roots
Let us talk about discriminant and the nature of roots in detail. The formulas for solving standard quadratic equations, i.e., ax2 + bx + c = 0, a ≠ 0. Thus, the roots (x) are:
x = (-b±√b2-4ac)/2a
The expression (b2 – 4ac) determines the nature of the roots of quadratic equations. The discriminant of the quadratic equation ax2 + bx + c = 0 is the term used to describe the expression. The discriminant is denoted by the symbol Δ, which is the Greek symbol for the letter D. The nature of the roots of a quadratic equation is defined by the discriminant. Therefore,
D/Δ = b2 – 4ac
Where D/Δ is discriminant.
The following are six major cases in which the type or nature of roots is determined. Look at the following basic formulae for solving quadratic equations once more.
α = (-b-√b2-4ac)/2a β = (-b+√b2-4ac)/2a
Case I: D/Δ > 0
When the discriminant (D) is positive, a, b, and c are real numbers and a ≠ 0, then the roots α and β of the quadratic equation (ax2+bx+ c = 0) will be real and unequal, and the curve meets at two different positions on the x-axis in graphical analysis.
Case II: D/Δ = 0
When the discriminant (D) is zero, a, b, and c are real numbers and a ≠ 0, then the roots α and β of the quadratic equation (ax2+bx+ c = 0) will be real and equal, and the curve intersects the x-axis at just one position in a visual depiction.
Case III: D/Δ < 0
When the discriminant (D) is negative, a, b and c are real numbers and a ≠ 0, then in this scenario, the roots α and β of the quadratic equation (ax2+bx+ c = 0) are imaginary or nonreal, and the curve does not meet the x-axis at any point in graphics depiction.
Case IV: D/Δ > 0 and perfect square
When the discriminant (D) is a positive and perfect square, a, b and c are real numbers and a ≠ 0, then in this scenario, the roots α and β of the quadratic equation (ax2+bx+ c = 0) are real, rational, and unequal.
Case V: D/Δ > 0 and not a perfect square
When the discriminant (D) is positive but not a perfect square, a, b, and c are real numbers and a ≠ 0, then the roots α and β of the quadratic equation (ax2+bx+ c = 0) will be real, irrational, and unequal and can be represented as a decimal. The roots α and β create a pair of irrational conjugates in this situation.
Case VI: D/Δ > 0 is a perfect square, and a or b is irrational
When the discriminant (D) is a perfect square but either a or b is irrational, a, b, and c are real numbers and a ≠ 0, then in this case, the roots α and β of the quadratic equation (ax2+bx+ c = 0) will be irrational.
Methods for determining the roots of quadratic equations
A value of an unknown element of the equation is the value of the quadratic equation’s roots. For example, since ax2 + bx + c =0, the value of x is the root of the quadratic equation. The following are some ways for calculating the roots of quadratic equations:
- Factorization methodology
- Completing the square approach
- Using the quadratic formula (Sridharacharya formula – Sridharacharya equation is given by ax2 + bx + c = 0, where a, b, c are real numbers and a ≠ 0)
Conclusion
The most crucial component of quadratic equation problems is finding the roots of a quadratic equation. If you don’t know how to calculate the roots of a quadratic equation, the equation cannot be solved. As a result, understanding all of the ideas connected to the roots of a quadratic equation is critical.
