# FORMULAS

An algebra is a collection of symbols,operators, and governing laws.

An algebra is a collection of symbols, operators, and governing laws.

# x³ – 4x² + 3x – 5x² + 20x-15 = 0

X^3-9x^2+23X-15=0

# SOLVING CUBICS GIVEN ONE REAL ROOT

## All cubic equation graphs cross the x axis at least once; in other words, they all have at least one real root. Take y=Ax^3+Bx^2+Cx+D=0. Assume p is the real root. y=(x-p)(fx^2+gx+h).We can divide x-p into y and we get y/(x-p)=(Ax^2+(B+Ap)x+(C+p(B+Ap)).We have the roots. Example:3x^3-5x^2+10x-24=0. A=f=3.B=-5.C=10.D=-24.We are given p=2. g=B+Ap=-5+3*2=1. h=C+pg=10+2*1=12=-D/p. We can compute the roots using the the quadratic formula:y=(fx^2+gx+h)(x-p)=(3x^2+x+12)(x-2).Therefore x=-1/6+-((1-4*3*12)^(1/2))/6=-1/6+-I((143)^1/2)/6. D’=g^2-4fh >0 implies real roots; D'<0 implies imaginary roots. Consider y=A(x-p)(x^2+gx/A+h/A). f=A. g=B+Ap h=C+pg. Example: y=x^3-2x^2-23x+60=0.A=1. B=-2. C=-23. We are given p=3.g=B+Ap=1. f=1.h=C+pg=-20.D’=g^2-4fh=1+4*1*20=81>0 and we have real roots: x=(-1+-(1-4*(-20))^1/2)/2=(-1+-9)/2=4,-5,and 3.

Therefore: y=(x-p)(Ax^2+gx+h)

y=A(x-p)(x^2+(B+Ap)x/A-D/Ap)