Solve the given problem that involves binomial theorem

[chwala]

Homework Statement

See attached

Relevant Equations

Binomial theorem

part (a)

##(4+3x)^{1.5} = 2^3+ 9x+ \left[\dfrac {1}{2} ⋅ \dfrac {3}{2} ⋅\dfrac {1}{2}⋅\dfrac {1}{2}⋅9x^2\right]+ ...##

##(4+3x)^{1.5}=8+9x+\dfrac {27}{16} x^2+...##part (b)

##x≠-\dfrac {4}{3}##part (c)

##(8+9x+\dfrac {27}{16} x^2+...)(1+ax)^2 = \dfrac{107}{16} x^2##

...

##8a^2+18a+\dfrac {27}{16}=\dfrac{107}{16}##

##8a^2+18a=\dfrac{80}{16}##

##128a^2+288a-80=0##

##8a^2+18a-5=0##

##a_1=0.25##

##a_2= -2.5##

Bingo!

Any other way welcome guys...

 

[pasmith]

The binomial expansion of [itex](1 + x)^\alpha[/itex] is only valid for [itex]|x| < 1[/itex]. [tex]
(4 + 3x)^{1.5} = 4^{1.5}\left(1 + \frac{3x}{4}\right)^{1.5}.[/tex]

 

[chwala]

The binomial expansion of [itex](1 + x)^\alpha[/itex] is only valid for [itex]|x| < 1[/itex]. [tex]
(4 + 3x)^{1.5} = 4^{1.5}\left(1 + \frac{3x}{4}\right)^{1.5}.[/tex]

I should have expressed my answer as,

##|x|<\dfrac{4}{3}##