Which statement is true about discriminant of quadratic equation $ax^2 + bx + c = 0$ ?

Question

Accepted Answer

The question asks about the discriminant of a quadratic equation $ ax^2 + bx + c = 0 $ and which statement is true regarding it. The discriminant is given by the expression $ b^2 - 4ac $. The correct answer is Option 4, which states "All of the above." Let's provide a detailed explanation to justify why this is correct and why each individual option is also correct, thus supporting Option 4.

1. **Two distinct real roots, if $ b^2 - 4ac > 0 $:**  
   This statement is true because when the discriminant $ b^2 - 4ac $ is greater than zero, it indicates that the quadratic equation has two distinct real roots. This is because the square root of a positive number is real and non-zero, leading to two different solutions when applying the quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $.

2. **Two equal real roots, if $ b^2 - 4ac = 0 $:**  
   This statement is also true. When the discriminant is exactly zero, the quadratic equation has exactly one real root, but it is counted as two equal roots. This is because the square root of zero is zero, and the quadratic formula simplifies to $ x = \frac{-b}{2a} $, providing a single solution that is repeated.

3. **No real roots, if $ b^2 - 4ac < 0 $:**  
   This statement is true as well. When the discriminant is less than zero, it indicates that the quadratic equation has no real roots. Instead, the roots are complex or imaginary because the square root of a negative number is not real. This results in complex conjugate roots.

4. **All of the above:**  
   Since each of the individual statements (Options 1, 2, and 3) is true under their respective conditions, Option 4, which states "All of the above," is indeed the correct answer. It encompasses all the possible scenarios for the discriminant of a quadratic equation, making it the most comprehensive and accurate choice.

In conclusion, Option 4 is correct because it accurately includes all the possible outcomes for the discriminant of a quadratic equation, each of which is correctly described by Options 1, 2, and 3.