need help with polynomials (1 Viewer)

[koizora]

1) find c if one integral roots of x^2-12x+c=0 double the other root<?xml:namespace prefix = o ns = "urn:schemas-microsoft-comfficeffice" /><o></o>
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2) what are possible values of b if the roots of x^2+bx-36 are integers?<o></o>
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3) for what values of k does the quadratic equation x^2 - kx + 2k-2= 0 has two roots differing by 1?<o></o>


4) the equation x^3 + 3Hx + L =0 has two equal roots.
Prove that L^2 = -4H^3

thank you

 

[blaze.bluetane]

just use the co-efficient to root relationships

1) let the roots be A and 2A

A+2A=12
A=4
2A=8

c=2A^2
c=32


2) x^2 + bx + 36 = 0, integral roots

let the roots be A and B

36= A x B

(A,B)=(4,9)(-4,-9)(3,12)(-3,-12)(6,6)(-6,-6)(2,18)(-2,-18)

Notice that (A,B) is the same as (B,A) [FYI]

So -b=A+B
b=13,-13,15,-15,12,-12,20,-20



3) x^2 - kx + 2k-2= 0

roots differ by 1

let the roots be A and A+1

A(A+1)=2k-2--------eqn 1

k=A+A+1
2A=k-1
A=(k-1)/2------------eqn 2


Solve simultaneously to find k (it is too messy to type)


4) the equation x^3 + 3Hx + L =0 has two equal roots.
Prove that L^2 = -4H^3

let the roots be A,A,B

A+A+B=0
B=-2A


A^2 + 2AB=3H
A(A+2B)=3H
but B=-2A
A(A-4A)=3H
-3A^2=3H
4A^6=-4H^3-----------eqn1

-L=A^2 . B
but B=-2A
L=2A^3
L^2=4A^6---------------eqn2

Therefore, equating equation 1 and equation 2,
L^2 = -4H^3



P.s. These questions are based on the relationships between roots and co-efficients section in the polynomials topic. If you don't understand this, then that is maybe why.