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## How To Tell The Nature Of Roots Of Quadratic Equations!

**Nature of Roots of Quadratic Equations**

Quadratic equations are degree two equations. When these are solved we get the solution as two values of the variable in them. The solution has many names, such as the root, the zero, and the value of the variable. KEY There are two values of variable and they can be real and imaginary. Mathematics students in class 10 to class 12 should know both types of solutions (roots). In this presentation I focus only on the actual roots.

There are three possibilities about the roots of a degree two equation. Since these equations have degree two, the variables involved in them have two values, but this is not always the case.

Sometimes there are two roots that are distinct and unique, sometimes the equation has both the same root and in other cases there is no solution to the equation. An unsolved equation means that there is no way to solve the equation to get the real value (real root) of the equation, and these types of equations can have imaginary roots.

There is a method for stating the nature of the roots of quadratic equations without solving the equation. This method involves finding the value of the discriminant (symbolized as D) for a quadratic equation.

The formula for finding the disciminant (D) is given below:

D = b² – 4ac

Where “D” stands for disciminant, “b” is the coefficient of the linear term, “a” is the coefficient of the quadratic term (the term with the square of the variable) and “c” is the constant term.

The differential is calculated using the above formula and the result is analyzed as given below:

1. When D > 0

In this case the equation has two distinct real roots.

2. When D = 0

In this case the equation has two common roots.

3. When D < 0

In this case there are no real roots for the equation.

For example; Consider we want to know the nature of the roots of the quadratic equation, “3x² – 5x + 3 = 0”

In this quadratic equation; a = 3, b = – 5 and c = 3. Use these values in the formula to find the discriminant for the equation shown below:

D = b² – 4ac

= (- 5)² – 4 (3) (3)

= 25 – 36

= – 11 <0

Therefore, D < 0 and the given equation has no real roots.

Finally, it can be said that differentiation is the key to predicting the nature of quadratic equations. Once the value of the discriminant is calculated, its formula nature can be used to estimate the root of the quadratic equation.

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