Quadratic Equations

For the quadratic equation $ax^2+bx+c=0$ ($a\ne0$), the quantity $\Delta=b^2-4ac$ is the discriminant. The two roots are $\dfrac{-b+\sqrt\Delta}{2a}$ and $\dfrac{-b-\sqrt\Delta}{2a}$, usually combined as

$\Delta=0$ exactly when the two roots coincide (a repeated/equal root). When $a,b,c$ are all real:

• $\Delta>0$ $\iff$ the roots are real and distinct;
• $\Delta<0$ $\iff$ the quadratic has no real roots (the two roots are a non-real complex conjugate pair — see §3.4.1). …