- #1

chwala

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- Homework Statement
- find the set of values of ##k## for which the equation

##3x^4+4x^3-12x^2+k =0## has 4 real roots

- Relevant Equations
- quartic polynomials

##x^2(3x^2+4x-12) +k=0##

##(3x^2+4x-12)= \frac{-k}{x^2}##

or

##(4x^3-12x^2)=-k-3x^4##

##4(3x^2-x^3)=3x^4+k##

##4x^2(3-x)= 3x^4+k##

or using turning points,

let ##f(x)= 3x^4+4x^3-12x^2+k##

it follows that,

##f'(x)=12x^3+12x^2-24x=0##

##12x(x^2+x-2)=0##

##12x(x-1)(x+2)=0## the turning points for this graph are at,

##x=0##, ## x=1## & ##x=-2##

or considering the first term and the constant, we have

##(ax+k=0)##, where ##a=3##, possible values for ## k = ±1, ±3...##

##(3x^2+4x-12)= \frac{-k}{x^2}##

or

##(4x^3-12x^2)=-k-3x^4##

##4(3x^2-x^3)=3x^4+k##

##4x^2(3-x)= 3x^4+k##

or using turning points,

let ##f(x)= 3x^4+4x^3-12x^2+k##

it follows that,

##f'(x)=12x^3+12x^2-24x=0##

##12x(x^2+x-2)=0##

##12x(x-1)(x+2)=0## the turning points for this graph are at,

##x=0##, ## x=1## & ##x=-2##

or considering the first term and the constant, we have

##(ax+k=0)##, where ##a=3##, possible values for ## k = ±1, ±3...##

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