What are the conditions under which a quadratic equation has real roots?

Answers:

• A quadratic equation of the form ( ax^2 + bx + c = 0 ) has real roots if its discriminant ( D ) is non-negative. The discriminant is given by the formula:

  [ D = b^2 - 4ac ]

  The conditions for the roots based on the value of the discriminant are as follows:

  1. Two distinct real roots: If ( D > 0 ), the quadratic equation has two distinct real roots.

  2. One real root (repeated root): If ( D = 0 ), the quadratic equation has exactly one real root, which is a repeated root.

  3. No real roots: If ( D < 0 ), the quadratic equation has no real roots, meaning the roots are complex (conjugate) numbers.

  Thus, the condition for a quadratic equation to have real roots is that the discriminant ( D ) must be greater than or equal to zero (( D \geq 0 )).