Question

If the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are real and equal, then:

• A. \( b^2 - 4ac < 0 \)
• B. \( b^2 - 4ac = 0 \)
• C. \( b^2 - 4ac > 0 \)
• D. \( a + b + c = 0 \)

Hint:

Remember: Discriminant zero means the quadratic has a repeated (equal) real root.

Answer 1

The Correct Option is B

Step 1: Define the discriminant
The discriminant is: \[ D = b^2 - 4ac \] The value of \( D \) determines the roots: 
- \( D > 0 \): Two distinct real roots 
- \( D = 0 \): Two real and equal roots (one repeated root) 
- \( D < 0 \): Two complex conjugate roots 

Step 2: Apply to the question 
The question states the roots are real and equal, so: \[ D = b^2 - 4ac = 0 \] 
Step 3: Check other options 
- Option (1): \( b^2 - 4ac < 0 \): Gives complex roots, incorrect. 
- Option (3): \( b^2 - 4ac > 0 \): Gives two distinct real roots, incorrect. 
- Option (4): \( a + b + c = 0 \): This is the sum of the coefficients, which equals the value of the quadratic at \( x = 1 \). It’s unrelated to the condition for equal roots unless specific values are given. Incorrect. 

Step 4: Conclusion 
The condition for real and equal roots is \( b^2 - 4ac = 0 \).