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 given by the formula: \[ D = b^2 - 4ac \] The nature of the roots is determined by the value of \( D \):
- If \( D > 0 \): There are two distinct real roots.
- If \( D = 0 \): There are two real and equal roots (a single repeated root).
- If \( D < 0 \): There are two complex conjugate roots.

Step 2: Apply to the Given Condition
The problem states that the roots are real and equal, which corresponds to the condition: \[ D = b^2 - 4ac = 0 \]

Step 3: Evaluate Alternative Scenarios
- Scenario (1): \( b^2 - 4ac < 0 \). This results in complex roots, which is incorrect for this problem.
- Scenario (3): \( b^2 - 4ac > 0 \). This results in two distinct real roots, which is incorrect.
- Scenario (4): \( a + b + c = 0 \). This equation represents the sum of the coefficients, which is equivalent to the value of the quadratic expression when \( x = 1 \). This condition is not directly related to having equal roots unless specific coefficients are provided. Therefore, it is incorrect.

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