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Question Number 79015    Answers: 0   Comments: 4

Rigorously over one month′s time, I developed a formula for general cubic. x^3 +ax^2 +bx+c=0 let x=((pt+q)/(t+1)) pq=m, p+q=s ________________________ m^2 {(a^2 +b)^2 −6a(ab−c)} +m{2(b^2 +ac)(a^2 +b)− 3(ab−c)(ab+3c)} +(b^2 +ac)^2 −6bc(ab−c)=0 ________________________ s=−(2/3){((m(a^2 +b)+b^2 +ac)/(ab−c))} +{(8/(27))[((m(a^2 +b)+b^2 +ac)/(ab−c))]^3 −8[((m^3 +bm^2 +acm+c^2 )/(ab−c))]}^(1/3) p,q = (s/2)±(√((s^2 /4)−m)) t=−(((3pq^2 +2apq+ap^2 +2bp+bq+3c))/((p^3 +ap^2 +bp+c))) x=((pt+q)/(t+1)) . (Please help checking..) (edited a digit 1 in place of 4)

$${Rigorously}\:{over}\:{one}\:{month}'{s} \\ $$$${time},\:{I}\:{developed}\:{a}\:{formula}\:{for} \\ $$$${general}\:{cubic}. \\ $$$${x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0} \\ $$$${let}\:\:{x}=\frac{{pt}+{q}}{{t}+\mathrm{1}} \\ $$$$\boldsymbol{{pq}}=\boldsymbol{{m}},\:\boldsymbol{{p}}+\boldsymbol{{q}}=\boldsymbol{{s}} \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$$\boldsymbol{{m}}^{\mathrm{2}} \left\{\left(\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}\right)^{\mathrm{2}} −\mathrm{6}\boldsymbol{{a}}\left(\boldsymbol{{ab}}−\boldsymbol{{c}}\right)\right\} \\ $$$$+\boldsymbol{{m}}\left\{\mathrm{2}\left(\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{ac}}\right)\left(\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}\right)−\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}\left(\boldsymbol{{ab}}−\boldsymbol{{c}}\right)\left(\boldsymbol{{ab}}+\mathrm{3}\boldsymbol{{c}}\right)\right\} \\ $$$$\:\:\:+\left(\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{ac}}\right)^{\mathrm{2}} −\mathrm{6}\boldsymbol{{bc}}\left(\boldsymbol{{ab}}−\boldsymbol{{c}}\right)=\mathrm{0} \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$$\boldsymbol{{s}}=−\frac{\mathrm{2}}{\mathrm{3}}\left\{\frac{\boldsymbol{{m}}\left(\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}\right)+\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{ac}}}{\boldsymbol{{ab}}−\boldsymbol{{c}}}\right\} \\ $$$$\:\:+\left\{\frac{\mathrm{8}}{\mathrm{27}}\left[\frac{\boldsymbol{{m}}\left(\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}\right)+\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{ac}}}{\boldsymbol{{ab}}−\boldsymbol{{c}}}\right]^{\mathrm{3}} \right. \\ $$$$\left.\:\:\:−\mathrm{8}\left[\frac{\boldsymbol{{m}}^{\mathrm{3}} +\boldsymbol{{bm}}^{\mathrm{2}} +\boldsymbol{{acm}}+\boldsymbol{{c}}^{\mathrm{2}} }{\boldsymbol{{ab}}−\boldsymbol{{c}}}\right]\right\}^{\mathrm{1}/\mathrm{3}} \\ $$$$\boldsymbol{{p}},\boldsymbol{{q}}\:=\:\frac{\boldsymbol{{s}}}{\mathrm{2}}\pm\sqrt{\frac{\boldsymbol{{s}}^{\mathrm{2}} }{\mathrm{4}}−\boldsymbol{{m}}} \\ $$$$\boldsymbol{{t}}=−\frac{\left(\mathrm{3}\boldsymbol{{pq}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{{apq}}+\boldsymbol{{ap}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{{bp}}+\boldsymbol{{bq}}+\mathrm{3}\boldsymbol{{c}}\right)}{\left(\boldsymbol{{p}}^{\mathrm{3}} +\boldsymbol{{ap}}^{\mathrm{2}} +\boldsymbol{{bp}}+\boldsymbol{{c}}\right)} \\ $$$$\boldsymbol{{x}}=\frac{\boldsymbol{{pt}}+\boldsymbol{{q}}}{\boldsymbol{{t}}+\mathrm{1}}\:. \\ $$$$\left({Please}\:{help}\:{checking}..\right) \\ $$$$\left({edited}\:{a}\:{digit}\:\mathrm{1}\:{in}\:{place}\:{of}\:\mathrm{4}\right) \\ $$

Question Number 79010    Answers: 1   Comments: 0

lim_(x→0) 3x ln(x) ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{3x}\:\mathrm{ln}\left(\mathrm{x}\right)\:? \\ $$

Question Number 79004    Answers: 2   Comments: 3

Question Number 79001    Answers: 0   Comments: 3

In a binomial distribution with mean 4 and the probability of success is 1/3, then the number of trials is

$${In}\:{a}\:{binomial}\:{distribution}\:{with}\:{mean}\:\mathrm{4}\:{and}\:{the}\:{probability}\:{of}\:{success}\:{is}\:\mathrm{1}/\mathrm{3},\:{then}\:{the}\:{number}\:{of}\:{trials}\:{is} \\ $$

Question Number 79000    Answers: 0   Comments: 2

The differential equation of x^2 +y^2 =25 is

$${The}\:{differential}\:{equation}\:{of}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{25}\:{is} \\ $$

Question Number 78999    Answers: 1   Comments: 2

if dy/dx+y=1+y/x,then the integrating factor (I,F) is

$${if}\:{dy}/{dx}+{y}=\mathrm{1}+{y}/{x},{then}\:{the}\:{integrating}\:{factor}\:\left({I},{F}\right)\:{is} \\ $$

Question Number 78998    Answers: 0   Comments: 0

The value of ∫_0 ^(pi) sin2xdx+2∫_0 ^(pi/2) cos2xdx

$${The}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{pi}} \mathrm{sin2xdx}+\mathrm{2}\int_{\mathrm{0}} ^{{pi}/\mathrm{2}} {cos}\mathrm{2}{xdx} \\ $$

Question Number 78979    Answers: 0   Comments: 4

the first n even numbers are known. if one number is deleted then the average of remaining numbers is ((2582)/(50)). find the value of deleted number.

$$\mathrm{the}\:\mathrm{first}\:\mathrm{n}\:\mathrm{even} \\ $$$$\mathrm{numbers}\:\mathrm{are}\:\mathrm{known}.\:\mathrm{if}\:\mathrm{one}\:\mathrm{number} \\ $$$$\mathrm{is}\:\mathrm{deleted}\:\mathrm{then}\:\mathrm{the}\:\mathrm{average} \\ $$$$\mathrm{of}\:\mathrm{remaining}\:\mathrm{numbers}\:\mathrm{is}\:\frac{\mathrm{2582}}{\mathrm{50}}. \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{deleted}\:\mathrm{number}. \\ $$

Question Number 78974    Answers: 3   Comments: 3

{_(3e^(2x) −y^2 =2 ) ^(e^x −y^2 =2lny−x) solve the system on R∗R

$$\left\{_{\mathrm{3e}^{\mathrm{2x}} −\mathrm{y}^{\mathrm{2}} =\mathrm{2}\:\:\:} ^{\mathrm{e}^{\mathrm{x}} −\mathrm{y}^{\mathrm{2}} =\mathrm{2lny}−\mathrm{x}} \right. \\ $$$$\mathrm{solve}\:\mathrm{the}\:\mathrm{system}\:\mathrm{on}\:\mathbb{R}\ast\mathbb{R} \\ $$

Question Number 78948    Answers: 1   Comments: 7

Prove by mathematical induction that. n^4 + 4n^2 + 11 is divisible by 16

$$\mathrm{Prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{that}. \\ $$$$\:\:\:\mathrm{n}^{\mathrm{4}} \:+\:\mathrm{4n}^{\mathrm{2}} \:+\:\mathrm{11}\:\:\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{16} \\ $$

Question Number 78946    Answers: 1   Comments: 2

solve cosx−(√3)sinx=1

$$\mathrm{solve} \\ $$$$\mathrm{cos}{x}−\sqrt{\mathrm{3}}{sinx}=\mathrm{1} \\ $$

Question Number 78944    Answers: 0   Comments: 1

Hello sirs i need your help to solve tan2x≥(√3) in [0;2π]. i want that you explain me if possible how we make graphic to determinate.

$$\mathrm{Hello}\:\mathrm{sirs}\:\mathrm{i}\:\mathrm{need}\:\mathrm{your}\:\mathrm{help}\:\mathrm{to}\:\mathrm{solve} \\ $$$$\mathrm{tan2}{x}\geqslant\sqrt{\mathrm{3}}\:\mathrm{in}\:\left[\mathrm{0};\mathrm{2}\pi\right]. \\ $$$$\mathrm{i}\:\mathrm{want}\:\mathrm{that}\:\mathrm{you}\:\mathrm{explain}\:\mathrm{me}\:\mathrm{if}\: \\ $$$$\mathrm{possible}\:\mathrm{how}\:\mathrm{we}\:\mathrm{make}\:\mathrm{graphic}\:\mathrm{to} \\ $$$$\mathrm{determinate}. \\ $$

Question Number 78941    Answers: 2   Comments: 0

Q.find the sum S=(2^3 /(2!))+(3^3 /(3!))+(4^3 /(4!))+.... then find Σ_(n=1) ^∞ (n^4 /(n!))

$${Q}.{find}\:{the}\:{sum} \\ $$$${S}=\frac{\mathrm{2}^{\mathrm{3}} }{\mathrm{2}!}+\frac{\mathrm{3}^{\mathrm{3}} }{\mathrm{3}!}+\frac{\mathrm{4}^{\mathrm{3}} }{\mathrm{4}!}+.... \\ $$$${then}\:{find}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{4}} }{{n}!} \\ $$

Question Number 78937    Answers: 1   Comments: 5

Question Number 78933    Answers: 0   Comments: 4

Question Number 78931    Answers: 1   Comments: 8

if x+(1/x)=a find x^n +(1/x^n )=?

$${if}\:{x}+\frac{\mathrm{1}}{{x}}={a} \\ $$$${find}\:{x}^{{n}} +\frac{\mathrm{1}}{{x}^{{n}} }=? \\ $$

Question Number 78897    Answers: 1   Comments: 11

Question Number 78894    Answers: 0   Comments: 1

if U=x^3 +y^3 +6 sinz+11 then u_z (1,0,pi) is

$${if}\:{U}={x}^{\mathrm{3}} +{y}^{\mathrm{3}} +\mathrm{6}\:{sinz}+\mathrm{11}\:{then}\:{u}_{{z}} \left(\mathrm{1},\mathrm{0},{pi}\right)\:{is} \\ $$

Question Number 78890    Answers: 0   Comments: 2

the local maximum value of f(x)=x^4 +32x

$${the}\:{local}\:{maximum}\:{value}\:{of}\:{f}\left({x}\right)={x}^{\mathrm{4}} +\mathrm{32}{x} \\ $$

Question Number 78889    Answers: 1   Comments: 0

the angle between the planes r^→ .(2i^→ +2j^→ +2k^→ )=4 and 4x−2y+2z=15 is

$${the}\:{angle}\:{between}\:{the}\:{planes}\:{r}^{\rightarrow} .\left(\mathrm{2}{i}^{\rightarrow} +\mathrm{2}{j}^{\rightarrow} +\mathrm{2}{k}^{\rightarrow} \right)=\mathrm{4}\:{and}\:\mathrm{4}{x}−\mathrm{2}{y}+\mathrm{2}{z}=\mathrm{15}\:{is} \\ $$

Question Number 78888    Answers: 1   Comments: 1

The roots of the equation (x−a)(x−b)=abx^2 are always

$$\mathrm{The}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\left({x}−{a}\right)\left({x}−{b}\right)={abx}^{\mathrm{2}} \:\mathrm{are}\:\mathrm{always} \\ $$

Question Number 78887    Answers: 0   Comments: 2

if teta=tan^(−1) (7) then sec teta is

$${if}\:{teta}=\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{7}\right)\:{then}\:\mathrm{sec}\:{teta}\:{is} \\ $$

Question Number 78885    Answers: 1   Comments: 0

The number of real numbers in [0,2pi] satisfying sin^(−1) x−2sin^2 x+1=0 is

$${The}\:{number}\:{of}\:{real}\:{numbers}\:{in}\:\left[\mathrm{0},\mathrm{2pi}\right]\:\mathrm{satisfying}\:\mathrm{sin}^{−\mathrm{1}} \mathrm{x}−\mathrm{2sin}^{\mathrm{2}} {x}+\mathrm{1}=\mathrm{0}\:{is} \\ $$

Question Number 78878    Answers: 1   Comments: 0

x^3 y′′′ − 3x^2 y′′+6xy′ −6y=x^4 ln(x),x>0

$${x}^{\mathrm{3}} \:{y}'''\:−\:\mathrm{3}{x}^{\mathrm{2}} {y}''+\mathrm{6}{xy}'\:−\mathrm{6}{y}={x}^{\mathrm{4}} \:{ln}\left({x}\right),{x}>\mathrm{0} \\ $$

Question Number 78877    Answers: 1   Comments: 6

if ((sin(A))/(sin(B)))=((sin(D))/(sin(C))) and A+B=C+D then prove that A+B=180 C+D=180

$${if}\:\:\:\frac{{sin}\left({A}\right)}{{sin}\left({B}\right)}=\frac{{sin}\left({D}\right)}{{sin}\left({C}\right)} \\ $$$${and}\:{A}+{B}={C}+{D} \\ $$$$ \\ $$$${then}\:{prove}\:{that}\:\: \\ $$$${A}+{B}=\mathrm{180} \\ $$$${C}+{D}=\mathrm{180} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 78880    Answers: 1   Comments: 0

x + (1/x) = 3 , x ∈ R (x^(2020) + (1/x^(2020) )) mod (10) = y y^2 − 1 = ?

$${x}\:+\:\frac{\mathrm{1}}{{x}}\:\:=\:\:\mathrm{3}\:\:\:,\:\:\:\:{x}\:\in\:\mathbb{R} \\ $$$$\left({x}^{\mathrm{2020}} \:+\:\frac{\mathrm{1}}{{x}^{\mathrm{2020}} }\right)\:\:{mod}\:\left(\mathrm{10}\right)\:\:=\:\:{y} \\ $$$${y}^{\mathrm{2}} \:−\:\mathrm{1}\:\:=\:\:? \\ $$

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