Short Answer
The nature of roots of a quadratic equation depends on the value of the discriminant, which is b² − 4ac. It helps us know whether the roots are real, equal, or imaginary.
If the discriminant is positive, the equation has two real and different roots. If it is zero, there is one real root. If it is negative, the equation has no real roots.
Detailed Explanation
Nature of roots using discriminant
A quadratic equation is written in the form:
The discriminant is given by:
The value of this expression helps us understand the nature of the roots, which means the type of solutions the equation has.
Cases of discriminant and roots
There are three main cases based on the value of the discriminant:
- When D > 0 (positive discriminant)
If the value of b² − 4ac is greater than zero, the quadratic equation has two real and different roots.
This means the equation has two distinct solutions. - When D = 0 (zero discriminant)
If the value of the discriminant is equal to zero, the equation has one real and equal root.
This means both roots are the same value. - When D < 0 (negative discriminant)
If the value of the discriminant is less than zero, the equation has no real roots.
The solutions are imaginary or complex numbers.
Understanding with examples
Let us understand each case with examples:
Example 1: D > 0
x² − 5x + 6 = 0
D = (−5)² − 4×1×6 = 25 − 24 = 1
Since D > 0, there are two real and different roots.
Example 2: D = 0
x² − 4x + 4 = 0
D = (−4)² − 4×1×4 = 16 − 16 = 0
Since D = 0, there is one real root.
Example 3: D < 0
x² + 2x + 5 = 0
D = 2² − 4×1×5 = 4 − 20 = −16
Since D < 0, there are no real roots.
Importance of discriminant
The discriminant is very useful because:
- It tells the type of roots without solving the equation fully
- It helps save time in calculations
- It helps in understanding the graph of the equation
- It is used in the quadratic formula
By just calculating one value, we can get important information about the equation.
Relation with graph
The nature of roots is closely related to the graph of the quadratic equation:
- When D > 0, the graph cuts the x-axis at two points
- When D = 0, the graph touches the x-axis at one point
- When D < 0, the graph does not touch the x-axis
So, the discriminant also tells us how the graph behaves.
Conclusion
The nature of roots of a quadratic equation depends on the value of the discriminant b² − 4ac. It helps us understand whether the equation has two real roots, one real root, or no real roots. This concept is very important for solving and analyzing quadratic equations easily.