Discriminant and Roots

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by $\Delta = b^2 - 4ac$. The value of the discriminant determines the nature of the roots:

• If $b^2 - 4ac > 0$, there are two distinct real roots.
• If $b^2 - 4ac = 0$, there is one repeated real root (equal roots).
• If $b^2 - 4ac < 0$, there are two complex conjugate roots.

Problems often ask to find a coefficient $k$ that results in a certain type of root. For example, for $3x^2 - x - 2k = 0$ to have equal roots, we set the discriminant to zero: $(-1)^2 - 4(3)(-2k) = 0 \implies 1 + 24k = 0 \implies k = -\frac{1}{24}$.