What are the methods for solving quadratic equations and what indicators predict that a quadratic function will have a complex solution?

Answer:  1) Methods: -  Quadratic formula. - Factorization. - Completing the square. 2) If the determinant is less than zero ([tex]D<0[/tex]) then there are two roots that are complex conjugates.Step-by-step explanation:  Methods: -  Quadratic formula Given the quadratic equation in Standard form [tex]ax^2+bx+c=0[/tex], you can solve it with the quadratic formula: [tex]x=\frac{-b\±\sqrt{b^2-4ac}}{2a}[/tex] - Factorization You must find two expression that when you multply them, you get the original quadratic equation. For example: [tex]x^2+6x+8=0[/tex] Find two number whose sum is 6 and whose product is 8. These are 2 and 4. Then: [tex](x+2)(x+4)=0[/tex] When you make the multiplication indicated in [tex](x+2)(x+4)=0[/tex], you obtain [tex]x^2+6x+8=0[/tex] - Completing the square Given the quadratic equation in Standard form [tex]ax^2+bx+c=0[/tex],, you must turn it into: [tex]a(x+d)^2+e=0[/tex] Where: [tex]d=\frac{b}{2a}\\\\e=c-\frac{b^2}{4a}[/tex] Once you get that form, you must solve for x. You can predict if the quadratic function will have a complex solution with the determinant: [tex]D=b^2-4ac[/tex] If [tex]D<0[/tex] then there are two roots that are complex conjugates.