Which are integer numbers #a,b# that verify simultaneously these equations: #2a^2+2b^2=200# and #3ab=144#?

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Answer 1 · 2017-06-23T13:42:15

The solutions are #(a,b)=(+-8,+-6)# or #(+-6,+-8)#

The first equation is

#2a^2+2b^2=200#, #=>#, #a^2+b^2=100#

The second equation is

#3ab=144#, #=>#, #ab=144/3=48#

#a=48/b#

Substututing this value in the first equation

#a^2+48^2/a^2=100#

#a^4+2304=100a^2#

#a^4-100a^2+2304=0#

Solving this quadratic equation in #a^2#

The discriminant is

#Delta=(-100)^2-4*(1)*(2304)=10000-9216=784#

As #Delta>0#, there are #2# real solutions

#a^2=(-(-100)+-sqrt784)/(2)=(100+-28)/2#

#a^2=128/2=64# or #a^2=72/2=36#

Therefore,

#a=+-8# or #a=+-6#

#b=48/a=48/8=+-6# or

#b=48/6=+-8#

The solutions are #(+-8,+-6)# or #(+-6,+-8)#