# Algebra Formulas part 3

Posted By on December 21, 2014

Solutions (roots):

 x1,2=−b±b2−4ac−−−−−−−√2a

If D=b24ac is the discriminant , then the roots are

1. real and unique if D>0

2. real and equal if D=0

3. complex conjugate if D<0

### Cubic Equation: x3+a1x2+a2x+a3=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 is the discriminant of the cubic equation, then:

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

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

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

### Quartic Equation:x4+a1x3+a2x2+a3x+a4=0

Let y1 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

#### Posted by Akash Kurup

Founder and C.E.O, World4Engineers Educationist and Entrepreneur by passion. Orator and blogger by hobby

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