Question

Given below are two statements
Statement I: A quadratic equation can have three distinct real roots.
Statement II: In quadratic equation \(ax^2 + bx + c = 0\), if \(b^2 - 4ac = 0\), then the quadratic equation has real and equal roots.
In the light of the above statements, choose the correct answer from the options given below:

• A. Both Statement I and Statement II are true
• B. Both Statement I and Statement II are false
• C. Statement I is true but Statement II is false
• D. Statement I is false but Statement II is true

Hint:

Quadratic equation: \(ax^2 + bx + c = 0\)
Discriminant \(D = b^2 - 4ac\)
\(D > 0\): Two real roots
\(D = 0\): Equal roots
\(D < 0\): No real roots

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question tests knowledge of quadratic equations and their roots.

Step 2: Detailed Explanation:
Statement I:
A quadratic equation has degree 2, so it can have at most 2 roots.
It can have 0, 1, or 2 real roots (depending on the discriminant).
Three distinct real roots are not possible for a quadratic equation.
This statement is false.
Statement II:
For the quadratic equation \(ax^2 + bx + c = 0\):
The discriminant is \(D = b^2 - 4ac\).
- If \(D > 0\): Two distinct real roots.
- If \(D = 0\): Real and equal roots (one repeated root).
- If \(D < 0\): No real roots (complex roots).
Thus, if \(b^2 - 4ac = 0\), the equation has real and equal roots.
This statement is true.

Step 3: Final Answer:
Thus, Statement I is false but Statement II is true.
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