Product of the Roots of a Quadratic

Multiply out a(x − x₁)(x − x₂) and the constant term lands as a·x₁x₂, so the two roots always multiply to c/a. Worked example: 4x² + 5x − 9 = 0 has P = −9/4 = −2.25, and the roots 1 and −9/4 confirm it. Paired with the sum relation S = −b/a, this pins a quadratic completely: any equation whose roots add to S and multiply to P is x² − Sx + P = 0 up to an overall scale factor.

The sign of P is a free diagnostic. Negative means the roots straddle zero (one positive, one negative) so a projectile's height equation with a negative constant term always has one physically meaningless negative time. Positive P with positive S means both roots are positive; positive P with negative S means both are negative. The trap: P = c/a, with no minus sign, unlike the sum. And remember these relations describe the roots you would get, real or complex: a quadratic with no real roots still has a perfectly real product c/a, because the two complex conjugates multiply to the square of their modulus.