Solutions of algebraic equations

Quadric Equation: ax2+bx+c=0$ax^2 + bx + c = 0$

Solutions (roots):

 x1,2=−b±b2−4ac−−−−−−−√2a

If D=b24ac$D = b^2 - 4ac$ is the discriminant , then the roots are

1. real and unique if D>0$D > 0$

2. real and equal if D=0$D = 0$

3. complex conjugate if D<0$D < 0$

Cubic Equation: x3+a1x2+a2x+a3=0$x^3 + a_1x^2 + a_2x + a_3 = 0$

Let

 QRST=3a2−a219=9a1a2−27a3−2a3154=R+Q3+R2−−−−−−−√−−−−−−−−−−−−√3=R−Q3+R2−−−−−−−√−−−−−−−−−−−−√3

Then solutions (roots) of the cubic equation are:

 x1x2x3=S+T−13a1=−12(S+T)−13a1+12i3√(S−T)=−12(S+T)−13a1−12i3√(S−T)

If D=Q3+R2$D = Q^3 + R^2$ is the discriminant of the cubic equation, then:

1. one root is real and two complex conjugate if D>0$D > 0$

2. all roots are real and at last two are equal if D=0$D=0$

3. all roots are real and unequal if D<0$D < 0$

Quartic Equation:x4+a1x3+a2x2+a3x+a4=0$x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 = 0$

Let y1$y_1$ be a real root of the cubic equation

 y3−a2y2+(a1a3−4a4)y+(4a2a4−a23−a21a4)=0

Then solutions of the quartic equation are the 4 roots of

 z2+12(a1±a21−4a2+4y1−−−−−−−−−−−−√)z+12(y1±y21−4a4−−−−−−−√)=0