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Question Number 42878 Answers: 1 Comments: 0

Question Number 42826 Answers: 0 Comments: 7

solving ax^4 +bx^3 +cx^2 +dx+e=0 (a≠0, b, c, d, e)∈Q special cases (easy to solve) ax^4 +e=0 solve at^2 +e=0 ⇒ x=±(√t_(1, 2) ) ax^4 +cx^2 +e=0 solve at^2 +ct+e=0 ⇒ x=±(√t_(1, 2) ) always try all factors of ±e because a(x−α)(x−β)(x−γ)(x−δ)=ax^4 +...+αβγδ ⇒ e=αβγδ next we must find the nature of the solutions 4 real solutions 2 real & 2 complex solutions 4 complex solutions a, b, c, d, e ∈Q ⇒ complex solutions always in conjugated pairs draw the function or calculate some values to find the number of real solutions divide by a x^4 +px^3 +qx^2 +rx+s=0 [p=(b/a) q=(c/a) r=(d/a) s=(e/a)] I′ll soon post some cases I′ve been able to solve as comments

$$\mathrm{solving} \\ $$$${ax}^{\mathrm{4}} +{bx}^{\mathrm{3}} +{cx}^{\mathrm{2}} +{dx}+{e}=\mathrm{0} \\ $$$$\left({a}\neq\mathrm{0},\:{b},\:{c},\:{d},\:{e}\right)\in\mathbb{Q} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{special}\:\mathrm{cases}\:\left(\mathrm{easy}\:\mathrm{to}\:\mathrm{solve}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:{ax}^{\mathrm{4}} +{e}=\mathrm{0}\:\mathrm{solve}\:{at}^{\mathrm{2}} +{e}=\mathrm{0}\:\Rightarrow\:{x}=\pm\sqrt{{t}_{\mathrm{1},\:\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:{ax}^{\mathrm{4}} +{cx}^{\mathrm{2}} +{e}=\mathrm{0}\:\mathrm{solve}\:{at}^{\mathrm{2}} +{ct}+{e}=\mathrm{0}\:\Rightarrow\:{x}=\pm\sqrt{{t}_{\mathrm{1},\:\mathrm{2}} } \\ $$$$ \\ $$$$\mathrm{always}\:\mathrm{try}\:\mathrm{all}\:\mathrm{factors}\:\mathrm{of}\:\pm{e} \\ $$$$\mathrm{because}\:{a}\left({x}−\alpha\right)\left({x}−\beta\right)\left({x}−\gamma\right)\left({x}−\delta\right)={ax}^{\mathrm{4}} +...+\alpha\beta\gamma\delta \\ $$$$\Rightarrow\:{e}=\alpha\beta\gamma\delta \\ $$$$ \\ $$$$\mathrm{next}\:\mathrm{we}\:\mathrm{must}\:\mathrm{find}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{solutions} \\ $$$$\mathrm{4}\:\mathrm{real}\:\mathrm{solutions} \\ $$$$\mathrm{2}\:\mathrm{real}\:\&\:\mathrm{2}\:\mathrm{complex}\:\mathrm{solutions} \\ $$$$\mathrm{4}\:\mathrm{complex}\:\mathrm{solutions} \\ $$$${a},\:{b},\:{c},\:{d},\:{e}\:\in\mathbb{Q}\:\Rightarrow\:\mathrm{complex}\:\mathrm{solutions}\:\mathrm{always}\:\mathrm{in} \\ $$$$\mathrm{conjugated}\:\mathrm{pairs} \\ $$$$\mathrm{draw}\:\mathrm{the}\:\mathrm{function}\:\mathrm{or}\:\mathrm{calculate}\:\mathrm{some}\:\mathrm{values} \\ $$$$\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{solutions} \\ $$$$ \\ $$$$\mathrm{divide}\:\mathrm{by}\:{a} \\ $$$${x}^{\mathrm{4}} +{px}^{\mathrm{3}} +{qx}^{\mathrm{2}} +{rx}+{s}=\mathrm{0} \\ $$$$\left[{p}=\frac{{b}}{{a}}\:\:{q}=\frac{{c}}{{a}}\:\:{r}=\frac{{d}}{{a}}\:\:{s}=\frac{{e}}{{a}}\right] \\ $$$$ \\ $$$$\mathrm{I}'\mathrm{ll}\:\mathrm{soon}\:\mathrm{post}\:\mathrm{some}\:\mathrm{cases}\:\mathrm{I}'\mathrm{ve}\:\mathrm{been}\:\mathrm{able}\:\mathrm{to}\:\mathrm{solve} \\ $$$$\mathrm{as}\:\mathrm{comments} \\ $$

Question Number 42775 Answers: 0 Comments: 2

my mother fell down and broke waist...so i ambusy for mother...age 85...i am now kolkata for mother

$${my}\:{mother}\:{fell}\:{down}\:{and}\:{broke}\:{waist}...{so}\:{i}\:{ambusy} \\ $$$${for}\:{mother}...{age}\:\mathrm{85}...{i}\:{am}\:{now}\:{kolkata}\:{for}\:{mother} \\ $$

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Question Number 42725 Answers: 1 Comments: 0

Question Number 42656 Answers: 1 Comments: 0

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Question Number 42550 Answers: 1 Comments: 3

2^(1/x) =(√x) Find x

$$\mathrm{2}^{\frac{\mathrm{1}}{{x}}} =\sqrt{{x}} \\ $$$${Find}\:{x} \\ $$

Question Number 42458 Answers: 0 Comments: 0

a boy is in front of a wall.The distance between boy and wall is 12ft.the boy is moving towards the wall in such a way that half the distance between the wall and him is crossed in one minutes.So find the time the boy reach to the wall

$${a}\:{boy}\:{is}\:{in}\:{front}\:{of}\:{a}\:{wall}.{The}\:{distance}\:{between} \\ $$$${boy}\:{and}\:{wall}\:{is}\:\mathrm{12}{ft}.{the}\:{boy}\:{is}\:{moving}\:{towards} \\ $$$${the}\:{wall}\:{in}\:{such}\:{a}\:{way}\:{that}\:\boldsymbol{{half}}\:\boldsymbol{{the}}\:\boldsymbol{{distance}} \\ $$$$\boldsymbol{{between}}\:\boldsymbol{{the}}\:\boldsymbol{{wall}}\:\boldsymbol{{and}}\:\boldsymbol{{him}}\:\boldsymbol{{is}}\:\boldsymbol{{crossed}}\:\boldsymbol{{in}}\:\boldsymbol{{one}} \\ $$$$\boldsymbol{{minutes}}.\boldsymbol{{S}}{o}\:{find}\:{the}\:{time}\:{the}\:{boy}\:{reach}\:{to}\:{the}\:{wall} \\ $$

Question Number 42331 Answers: 1 Comments: 0

Question Number 42330 Answers: 0 Comments: 0

A linear function f(x)=ax + b transforms X={1,2,3,5,9,11} into Y,so that f(5)=13 and f(1)=5 Calculate the mean and Variance of X and Y.

$${A}\:{linear}\:{function}\:{f}\left({x}\right)={ax}\:+\:{b}\:{transforms}\:{X}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{9},\mathrm{11}\right\} \\ $$$${into}\:{Y},{so}\:{that}\:{f}\left(\mathrm{5}\right)=\mathrm{13}\:{and}\:{f}\left(\mathrm{1}\right)=\mathrm{5} \\ $$$${Calculate}\:{the}\:{mean}\:{and}\:{Variance}\:{of}\:{X}\:{and}\:{Y}. \\ $$

Question Number 42299 Answers: 1 Comments: 0

The set X and Y have five elementseach . Given that ΣX=25,ΣY=55,ΣX^2 =165 and ΣY^2 =765 and a linear Function y= px + q tranforms the set X into the set Y,where p and q are positive constants. a) Find the mean and Variance of X and Y hence,or otherwise , b)find the values of p and q.

$${The}\:{set}\:{X}\:{and}\:{Y}\:{have}\:{five}\:{elementseach}\:. \\ $$$${Given}\:{that}\:\Sigma{X}=\mathrm{25},\Sigma{Y}=\mathrm{55},\Sigma{X}^{\mathrm{2}} =\mathrm{165}\:{and}\:\Sigma{Y}^{\mathrm{2}} =\mathrm{765} \\ $$$${and}\:{a}\:{linear}\:{Function}\:{y}=\:{px}\:+\:{q}\:\:{tranforms}\:{the}\:{set}\:{X}\:{into}\:{the}\:{set}\: \\ $$$${Y},{where}\:{p}\:{and}\:{q}\:{are}\:{positive}\:{constants}. \\ $$$$\left.{a}\right)\:{Find}\:{the}\:{mean}\:{and}\:{Variance}\:{of}\:{X}\:{and}\:{Y} \\ $$$${hence},{or}\:{otherwise}\:, \\ $$$$\left.{b}\right){find}\:{the}\:{values}\:{of}\:{p}\:{and}\:{q}. \\ $$

Question Number 42287 Answers: 1 Comments: 0

Your family will be attending a family reunion at a particular beach resort. To avoid hassle, you consider renting a car that charges a flat rate of P2 000 plus P150 per kilometer. Write a piecewise function that model the situation.

$${Your}\:{family}\:{will}\:{be}\:{attending}\:{a}\: \\ $$$${family}\:{reunion}\:{at}\:{a}\:{particular}\:{beach} \\ $$$${resort}.\:{To}\:{avoid}\:{hassle},\:{you}\:{consider} \\ $$$${renting}\:{a}\:{car}\:{that}\:{charges}\:{a}\:{flat}\:{rate} \\ $$$${of}\:{P}\mathrm{2}\:\mathrm{000}\:{plus}\:{P}\mathrm{150}\:{per}\:{kilometer}.\: \\ $$$${Write}\:{a}\:{piecewise}\:{function}\:{that}\: \\ $$$${model}\:{the}\:{situation}. \\ $$

Question Number 42264 Answers: 1 Comments: 0

Thd local barangay recieved a budget of 150000 to provide medical check−ups for the children in the barangay. Write an equation representing the relationship of the alloted money per child.

$${Thd}\:{local}\:{barangay}\:{recieved}\:{a}\:{budget}\:{of}\:\mathrm{150000}\:{to}\:{provide}\:{medical}\:{check}−{ups}\:{for}\:{the}\:{children}\:{in}\:{the}\:{barangay}.\:{Write}\:{an}\:{equation}\:{representing}\:{the}\:{relationship}\:{of}\:{the}\:{alloted}\:{money}\:{per}\:{child}. \\ $$

Question Number 42224 Answers: 0 Comments: 5

a + b + c = 180 a,b,c ∈ N number of triplets possible (a,b,c) for the above equation are ? ( the order of a,b,c doesn′t matter)

$${a}\:+\:{b}\:+\:{c}\:=\:\mathrm{180}\: \\ $$$${a},{b},{c}\:\in\:\mathbb{N} \\ $$$${number}\:{of}\:{triplets}\:{possible} \\ $$$$\left({a},{b},{c}\right)\:{for}\:{the}\:{above} \\ $$$${equation}\:{are}\:? \\ $$$$\left(\:{the}\:{order}\:{of}\:{a},{b},{c}\:\:{doesn}'{t}\right. \\ $$$$\left.{matter}\right) \\ $$

Question Number 42207 Answers: 1 Comments: 0

Find the equation of tangent and normal to the curve y given by y = x^3 + 3x^2 + 7 .

$${Find}\:{the}\:{equation}\:{of}\:{tangent}\:{and}\:{normal}\:{to}\:\:{the}\:{curve}\:{y} \\ $$$${given}\:{by}\:\:\:{y}\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{7}\:. \\ $$

Question Number 42145 Answers: 1 Comments: 2