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1800-102-2727A quadratic equation is an equation of degree 2 in the form ax²+bx+c = 0, where a is not equal to 0. The value of x in this equation is called the roots of the quadratic equation. There are only two roots in a quadratic equation. The nature of these roots can be real and imaginary.
Value of discriminant | Nature of roots | |
D > 0 | Real, distinct | |
D is a perfect square | Rational roots | |
D is not a perfect square | Irrational roots | |
D = 0 | Real, equal | |
D < 0 | Complex, distinct (a pair of complex conjugates) |
Where D is called the discriminant of the root and is equal to b² – 4ac.
The universal equation to find roots of a quadratic equation ax² + bx + c = 0 is given by:
1. Factorization method: - It is the simplest method to find the roots of a quadratic equation. It is not applicable to all forms of the quadratic equation but is suitable for simple forms.
Example
Find the roots of the quadratic equation x² – 6x + 5 = 0 by factorization method.
Solution
By trial and error, we need to find the factors that make 6x and can be taken common from the terms x² and 5.
The above equation can be written as x² – 5x – x + 5 = 0
This implies, x (x-5) – 1 (x-5) = 0
This gives, (x-5) (x-1) = 0
We get the roots as x = 1 or x = 5.
2. Quadratic formula: - We can use the universal quadratic formula to find the roots of every quadratic equation.
Example
Find the roots of the equation x² – 6x + 5 = 0 from the quadratic formula.
Solution
Compare the given equation with the equation ax² + bx + c = 0.
We get, a = 1, b = -6, c = 5
By putting these values in the universal equation, we get
We get, x = 5 or x = 1, which are the roots of the quadratic equation x² – 6x + 5.