Discriminant of a quadratic equation, ax2 + bx + c = 0, can be found by finding the values using b2 – 4ac.
Now, on solving b2 – 4ac, you will get following three types of solutions of a quadratic equation:
1) b2 – 4ac > 0
2) b2 – 4ac < 0
3) b2 – 4ac = 0
In 1st condition, b2 – 4ac > 0, the given quadratic equation has two distinct real roots.
In 2nd condition, b2 – 4ac < 0, the given quadratic equation has no real roots.
In 3rd condition, b2 – 4ac = 0, the given quadratic equation has two equal real roots.
Now, let’s understand all the three conditions with the help of following examples.
1st Condition : b2 – 4ac > 0, the given quadratic equation has two distinct real roots.Consider the following quadratic equation:
2x2 -5x + 3 = 0
You can use any method to find the roots of the given quadratic equation. In this website, we have explained two methods of finding roots of quadratic equations:
1) Factorization method - Splitting middle term
| 2) Quadratic Equation: x = | -b + √ ( b 2 - 4ac )
2a |
Now, on applying any of the above method, we get roots of given quadratic equation, 2x2 -5x + 3 = 0, as shown below:
x = 3/2 and x = 1 are the roots of given quadratic equation.
So, you can see that the roots of x are distinct as well as real numbers.
Hence, this satisfy 1st condition which say that if b2 – 4ac > 0, then the given quadratic equation has two distinct real roots.
2nd Condition : To understand condition 2, b2 – 4ac < 0, the given quadratic equation has no real roots.Consider the following quadratic equation:
5x2 -6x - 2 = 0
Use quadratic equation to find the roots of the given quadratic equation and roots are as follows:
| x = | 3 + √ ( 19 )
5 | and | x = | 3 - √ ( 19 )
5 |
So, you can see that the roots of x are unreal numbers.
Hence, this satisfy 2nd condition which says that if b2 – 4ac < 0, then the given quadratic equation has no real root
3rd Condition : To understand condition 3, b2 – 4ac = 0, the given quadratic equation has two equal real roots. Consider the following quadratic equation:
x2 +2x +1 = 0
Here also, you can use any method to find the roots of the given quadratic equation. In this website, we have explained two methods of finding roots of quadratic equations:
1) Factorization method - Splitting middle term
| 2) Quadratic Equation: x = | -b + √ ( b 2 - 4ac )
2a |
Now, on applying any of the above method, we get roots of given quadratic equation, 2x2 -5x + 3 = 0, as shown below:
x = -1 and x = -1 are the roots of given quadratic equation.
So, you can see that the roots of x are equal numbers.
Hence, this satisfy 3rd condition which says that if b2 – 4ac = 0, then the given quadratic equation has two equal real roots.
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