Test for Quadratic Equations - Nature of Roots

Quadratic equations are polynomial equations of degree 2, which can be expressed in the form ax^2 + bx + c = 0, where a, b, and c are constants and 'a' is not equal to zero. The nature of the roots of a quadratic equation can be determined by analyzing the discriminant (D), which is given by the formula D = b^2 - 4ac.

The discriminant helps us understand the different types of roots a quadratic equation can have. Let's discuss the nature of roots and how to test them through an example.

Example:
Consider the quadratic equation 2x^2 + 5x + 3 = 0. We can find the nature of its roots by calculating the discriminant.

Step 1: Calculate the discriminant (D)
In this case, a = 2, b = 5, and c = 3. Substituting these values into the formula D = b^2 - 4ac, we get:
D = (5)^2 - 4(2)(3)
D = 25 - 24
D = 1

Step 2: Determine the nature of roots based on the value of D
The nature of roots can be determined as follows:

1. If D > 0, the quadratic equation has two distinct real roots.
2. If D = 0, the quadratic equation has two identical real roots.
3. If D < 0,="" the="" quadratic="" equation="" has="" no="" real="" roots.="" it="" only="" has="" complex="" />

Step 3: Analyze the value of D in our example
In our example, D = 1, which is greater than zero. Therefore, the quadratic equation 2x^2 + 5x + 3 = 0 has two distinct real roots.

Conclusion: The quadratic equation 2x^2 + 5x + 3 = 0 has two distinct real roots.

Summary:
To test the nature of roots of a quadratic equation, follow these steps:
1. Calculate the discriminant (D) using the formula D = b^2 - 4ac.
2. Analyze the value of D:
- If D > 0, the equation has two distinct real roots.
- If D = 0, the equation has two identical real roots.
- If D < 0,="" the="" equation="" has="" no="" real="" roots,="" only="" complex="" />

Remember that the discriminant helps us understand the nature of the roots and the number of real solutions a quadratic equation possesses.

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