Question

If the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are real and distinct, then which of the following conditions must be true?

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

Hint:

To determine the nature of the roots of a quadratic equation, use the discriminant \( \Delta = b^2 - 4ac \). - If \( \Delta > 0 \), the roots are real and distinct. - If \( \Delta = 0 \), the roots are real and equal. - If \( \Delta < 0 \), the roots are complex.

Answer 1

The Correct Option is A

For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is a key factor in determining the nature of the roots. The discriminant is given by the formula: \[ \Delta = b^2 - 4ac \] The discriminant provides information about the nature of the roots of the quadratic equation as follows: 1. If \( \Delta > 0 \), the quadratic equation has real and distinct roots. This means the equation has two distinct real solutions. 2. If \( \Delta = 0 \), the quadratic equation has real and equal roots. In this case, both roots are the same and real. 3. If \( \Delta < 0 \), the quadratic equation has complex (imaginary) roots. This means the roots are not real but involve imaginary numbers. ### Step-by-step explanation: We are asked for the condition where the roots are real and distinct. This condition corresponds to the case when \( \Delta > 0 \). Therefore, we need to have: \[ b^2 - 4ac > 0 \] - Explanation of the options: - Option (a) \( b^2 - 4ac > 0 \): This is the correct answer. When the discriminant is greater than zero, the quadratic equation has real and distinct roots. - Option (b) \( b^2 - 4ac = 0 \): This would give real and equal roots, not distinct roots. So, this condition is incorrect for the case of distinct roots. - Option (c) \( b^2 - 4ac < 0 \): This condition would lead to complex roots, so it is also incorrect for distinct roots. - Option (d) \( a + b + c = 0 \): This condition is not related to the nature of the roots in a general quadratic equation. This is an independent condition and does not guarantee real or distinct roots. Thus, the condition that ensures real and distinct roots is \( b^2 - 4ac > 0 \), which corresponds to Option (a). Therefore, the correct answer is: \[ \boxed{(a) \, b^2 - 4ac > 0} \]