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How To Tell The Nature Of Roots Of Quadratic Equations!

Nature of roots of quadratic equations

Quadratic equations are equations of degree two. When these are solved, we get the solution as two values ​​of the variable they contain. Solutions have many names, such as roots, zeros, and variable value. The key is that there are two values ​​of the variable and they can be real and imaginary. In grades 10 to 12, math students need to know both types of solutions (roots). In this presentation, I focus only on true roots.

There are three possibilities regarding the roots of second degree equations. Since the degree of these equations is two, they have two values ​​of the variable they contain, but this is not always the case.

Sometimes there are two distinct and unique roots, sometimes an equation has the same roots, and other times 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 to these types of equations.

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

The formula to find the discriminant (D) is given below:

D = b² – 4ac

Where “D” means 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 distinct 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 c = 3. Use these values ​​in the formula to find the discriminant for 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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