Nature of roots of quadratic equations
Quadratic equations are degree two equations. When these are solved we get the solution in the form of two values of the variable in them. Solutions have many names, such as, roots, zeroes and value of the variable. The key is there are two values of the variable and they can be real and imaginary. In grade ten to grade twelve math students need to know both kind of solutions (roots). In this presentation I am focusing on real roots only.
There are three possibilities about the roots of the degree two equations. As the degree of these equations is two, they have two values of the variable contained in them, but that is not the case all the times.
Some times there are two roots which are distinct and unique, some of the times an equation has both the same roots and in other cases there is no solution to the equation. No solution to equation means there is no such a way to solve the equation to get real value (real roots) of the equation and there could be imaginary roots to these kinds of equations.
There is a method to tell the nature of roots of quadratic equations without solving the equation. This method involves finding the value of discriminant (D as symbol) for the quadratic equation.
The formula to find 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 (term with square of the variable) and "c" is the constant term.
Disciminant is calculated using the above formula and the result is analyzed as given 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 we want to know the nature of 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
Hence, D < 0 and the given equation have no real roots.
Finally, it can be said that the discriminant is the key to predict the nature of the quadratic equations. Once the value of discriminant is calculated using its formula nature of the roots of a quadratic equation can be predicted.
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