solutions of quadratic equations

$a{x}^2+bx+c=0$ discriminant $\Delta =b^2-4ac$ $\Delta >0$ two distinct real solutions $\Delta =0$ one real solution $\Delta <0$ no real solutions

we can use complex numbers to get the real solutions though!

e.g. solve $x^2-5x+7=0$ $\Delta =(-5{)}^2-4\times 1\times 7=-3$ so there are no real solutions. nevertheless, the quadratic formula yields x= \frac{5\pm \sqrt{ -3 }}{2}$$$$=\frac{5}{2}\pm \frac{\sqrt{ 3 }i}{2} therefore there are two complex solutions: $x=\frac{5}{2}+\frac{\sqrt{3}}{2}i\text{ or }x=\frac{5}{2}-\frac{\sqrt{3}}{2}i$

two complex numbers $z$ and $w$ are equal if Re(z) = Re(w) and Im(z) = Im(w)

• you can also linearly factor quadratic polynomials