1.1.2—Quadratic discriminant

For ax²+bx+c=0, the discriminant Δ=b²−4ac. Δ>0 gives two distinct real roots, Δ=0 one repeated real root, and Δ<0 no real roots.

Use the discriminant when the question asks how many roots, a tangent condition or a parameter range. It is often cleaner than applying the quadratic formula twice.

For x²−4x+k=0, Δ=16−4k. Two real roots require k<4; tangency occurs at k=4.

Δ>0 does not mean both roots are positive; root signs require additional information such as sum and product.