Discriminant - Quadratic Formula

Quadratic Formula

The quadratic polynomial P(x) = ax2 + bx + c has discriminant

which is the quantity under the square root sign in the quadratic formula. For real numbers a, b, c, one has:

• When Δ > 0, P(x) has two distinct real roots

and its graph crosses the x-axis twice.

• When Δ = 0, P(x) has two coincident real roots

and its graph is tangent to the x-axis.

• When Δ < 0, P(x) has no real roots, and its graph lies strictly above or below the x-axis. The polynomial has two distinct complex roots

An alternative way to understand the discriminant of a quadratic is to use the characterization as "zero if and only if the polynomial has a repeated root". In that case the polynomial is The coefficients then satisfy so and a monic quadratic has a repeated root if and only if this is the case, in which case the root is Putting both terms on one side and including a leading coefficient yields