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

**Nature of the roots of equations of the second degree**

Quadratic equations are equations of degree two. When they are solved, we get the solution in the form of two values of the variable there. Solutions have many names, including roots, zeros, and value of the variable. The key is that there are two values of the variable and they can be real and imaginary. In grades ten through twelve, math students must know both types of solutions (roots). In this presentation I will focus only on the real roots.

There are three possibilities about the roots of quadratic equations. Since the degree of these equations is two, they have two values of the variable contained in them, but this is not always the case.

Sometimes there are two roots that are distinct and unique, sometimes an equation has both roots, and in other cases there is no solution to the equation. No solution to the equation means that there is no way to solve the equation to get the real value (real roots) of the equation and there could be imaginary roots for this type of equation.

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

The formula to find the discriminant (D) is as follows:

D = b² – 4ac

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

The discriminant is calculated using the above formula and the result is analyzed as shown below:

1. When D > 0

In this case there are two different real roots of the equation.

2. When D = 0

In this case there are two equal roots for the equation.

3. When D < 0

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

For example; consider that 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 ic = 3. Use these values in the formula to find the discriminant of the given equation as 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 the discriminant is the key to predicting the nature of quadratic equations. Once the value of the discriminant is calculated using its formula, the nature of the roots of a quadratic equation can be predicted.

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