Math question - exam on wed please help!?

Question

i downloaded a previous exam paper off the nzqa site, and i don't understand what the excellence question means/ how to do it. here it is:

Find the values of m for which one root of the equation 4x^2=mx-5 is three times the other root.

Never heard anything like it before. Any ideas?

Answer 1

An explicit formula is derived for the logarithmic Mahler measure $m(P)$ of $P(x,y) = p(x)y - q(x)$, where $p(x)$ and $q(x)$ are cyclotomic. This is used to find many examples of such polynomials for which $m(P)$ is rationally related to the Dedekind zeta value $\zeta_F (2)$ for certain quadratic and quartic fields.

Answer 2

4x^2-mx+5=0

any eqn of form ax^2+bx+c

has two roots the sum of two roots is always = -b/a

and the product of the 2 roots are always = c/a

remember that!!

say one root is ''r'' and other root is ''t'' then t=3r

also here sum of roots=t+r = m/4

product of roots=rt=5/4

=>3r+r=m/4

3r^2=5/4

=>r=m/16 and r^2=5/12

=>r^2=m^2/256  and r^2=5/12

=>m^2/256=5/12

=>m^2=320/3

=>m=+or-root 320/3

Answer 3

4x^2-mx+5=0

Suppose the roots are u and 3u then we have:

u+3u=4u=m/4 and u(3u)=3u^2=5/4

4u=m/4 then u=m/16

3u^2=5/4 then 3(m/16)^2=5/4

3m^2/256=5/4

12m^2=5(256) or m^2=5(256)/12

m=16(5 sqrt)/2(3sqrt)

m=8(15sqrt)/3

Answer 4

refer to your text