Menu Close

Solve-in-R-x-2-4x-40x-2-32x-16-




Question Number 156809 by MathSh last updated on 15/Oct/21
Solve in R  x^2  + 4x = (√(40x^2  + 32x - 16))
SolveinRx2+4x=40x2+32x16
Commented by MathSh last updated on 15/Oct/21
Yes dear Ser
YesdearSer
Answered by Rasheed.Sindhi last updated on 15/Oct/21
 x(x+4)= (√(40x^2  + 32x + 16))  x^2 (x+4)^2 =40x^2  + 32x + 16  x^4 +8x^3 +16x^2 −40x^2  − 32x − 16=0  x^4 +8x^3 −24x^2  − 32x − 16=0  x=3.1850,−10.0814 (Using calculator)  You can see exact results in an  advanced calculator by yourself.  They′re too complicated!
x(x+4)=40x2+32x+16x2(x+4)2=40x2+32x+16x4+8x3+16x240x232x16=0x4+8x324x232x16=0x=3.1850,10.0814(Usingcalculator)Youcanseeexactresultsinanadvancedcalculatorbyyourself.Theyretoocomplicated!
Answered by MJS_new last updated on 16/Oct/21
x^2 +4x=(√(40x^2 +32x−16))  squaring & transforming  x^4 +8x^3 −24x^2 −32x+16=0  (x^2 +4(1−(√2))x−4)(x^2 +4(1+(√2))x−4)=0  of the 4 solutions 3 fit the given equation  x_1 =−2+2(√2)+2(√(4−2(√2)))  x_(2, 3) =−2−2(√2)±2(√(4+2(√2)))
x2+4x=40x2+32x16squaring&transformingx4+8x324x232x+16=0(x2+4(12)x4)(x2+4(1+2)x4)=0ofthe4solutions3fitthegivenequationx1=2+22+2422x2,3=222±24+22
Commented by MJS_new last updated on 16/Oct/21
yes. if the equation for a has only 1 real  solution (Cardano) or if it has 3 real solutions  of trigonometric shape we cannot use them.  we must hope for easy solutions ∈Q to be  found within the factors of ±n
yes.iftheequationforahasonly1realsolution(Cardano)orifithas3realsolutionsoftrigonometricshapewecannotusethem.wemusthopeforeasysolutionsQtobefoundwithinthefactorsof±n
Commented by MathSh last updated on 16/Oct/21
Very nice dear Ser thanks
VerynicedearSerthanks
Commented by Rasheed.Sindhi last updated on 16/Oct/21
Thanks for precious knowledge!
Thanksforpreciousknowledge!
Commented by MJS_new last updated on 16/Oct/21
you′re welcome
yourewelcome
Commented by Rasheed.Sindhi last updated on 16/Oct/21
Sir, how did you factor the equation?
Sir,howdidyoufactortheequation?
Commented by MJS_new last updated on 16/Oct/21
the method I use:  x^4 +ax^3 +bx^2 +cx+d=0  let x=t−(a/4) to get  t^4 +pt^2 +qt+r=0  we want to find α, β, γ  to get  (t^2 −αt−β)(t^2 +αt−γ)=0  t^4 −(α^2 +β+γ)t^2 +α(γ−β)t+βγ=0  ⇒  { ((p=−(α^2 +β+γ))),((q=α(γ−β))),((r=βγ)) :}  solve the 1^(st)  & 2^(nd)  for β & γ  inserting in the 3^(rd)  & transforming we get  α^6 +jα^4 +kα^2 +l=0  let α=(√(a−(j/3))) to get  a^3 +ma+n=0  if this has an “easy to handle” solution we  succeed, otherwise the given equation has  no useable exact solution.
themethodIuse:x4+ax3+bx2+cx+d=0letx=ta4togett4+pt2+qt+r=0wewanttofindα,β,γtoget(t2αtβ)(t2+αtγ)=0t4(α2+β+γ)t2+α(γβ)t+βγ=0{p=(α2+β+γ)q=α(γβ)r=βγsolvethe1st&2ndforβ&γinsertinginthe3rd&transformingwegetα6+jα4+kα2+l=0letα=aj3togeta3+ma+n=0ifthishasaneasytohandlesolutionwesucceed,otherwisethegivenequationhasnouseableexactsolution.
Commented by MJS_new last updated on 16/Oct/21
x^2 +4x=(√(40x^2 +32x+16)) has no useable exact solution
x2+4x=40x2+32x+16hasnouseableexactsolution
Commented by Rasheed.Sindhi last updated on 16/Oct/21
Thanks a lot sir! I used a calculator  and it gave me terrible answers  in exact form!
Thanksalotsir!Iusedacalculatoranditgavemeterribleanswersinexactform!

Leave a Reply

Your email address will not be published. Required fields are marked *