Discriminant

The discriminant is the expression b² − 4ac, taken from the coefficients of a quadratic equation in standard form ax² + bx + c = 0. Its sign, without doing any further work, tells you how many real solutions the equation has.

A positive discriminant means two distinct real solutions. A discriminant of exactly zero means one repeated real solution. A negative discriminant means no real solutions, only complex ones. This makes it a fast diagnostic check before committing to the full quadratic formula.

Discriminant = b² − 4ac

• If discriminant > 0: two distinct real solutions
• If discriminant = 0: one repeated real solution
• If discriminant < 0: no real solutions (two complex solutions)

For the equation 2x² + 5x − 3 = 0, where a = 2, b = 5, c = −3:

• Discriminant = 5² − 4(2)(−3) = 25 + 24 = 49

Since 49 is positive, this equation has two distinct real solutions, confirmed by solving it fully: x = 1/2 and x = −3.

• A perfect square discriminant means rational solutions. If the discriminant works out to a perfect square (like 49, giving √49 = 7), the solutions will be clean rational numbers rather than messy decimals.
• Zero discriminant corresponds to a parabola's vertex touching the x-axis. Graphically, this is the one case where the parabola just grazes the axis instead of crossing it twice or missing it entirely.
• Negative discriminant doesn't mean the equation is wrong. It simply means the solutions are complex numbers, which is a completely valid mathematical outcome, just not a real-number one.
• Checking the discriminant first can save solving time. If you only need to know how many real solutions exist, not the actual values, the discriminant alone answers that without applying the full quadratic formula.
• Works for any quadratic in standard form. As long as the equation is written as ax² + bx + c = 0 with a ≠ 0, the discriminant formula applies directly.