MHB What are the solutions for (a,b) if 5a^2+5ab+5b^2=7a+14b?

[Albert1]

$a,b$ are all integers

(1) $if : \,\, 5a^2+5ab+5b^2=7a+14b$

please find all solutions of $(a,b)$

 

[MarkFL]

My solution:

Spoiler

Applying the quadratic formula, we find:

$$b=\frac{(14-5a)\pm\sqrt{14^2-3(5a)^2}}{10}$$

Now we require:

$$14^2-3(5a)^2\ge0$$

which implies:

$$a\in\{-1,0,1\}$$

Case 1: $$a=-1$$

$$b=\frac{19\pm11}{10}$$

The integral value is $$b=3$$

Hence, $(a,b)=(-1,3)$ is a solution.

Case 2: $$a=0$$

$$b=\frac{14\pm14}{10}$$

The integral value is $$b=0$$

Hence, $(a,b)=(0,0)$ is a solution.

Case 3: $$a=1$$

$$b=\frac{9\pm11}{10}$$

The integral value is $$b=2$$

Hence, $(a,b)=(1,2)$ is a solution.

 

[Albert1]

My solution:

Spoiler

Applying the quadratic formula, we find:

$$b=\frac{(14-5a)\pm\sqrt{14^2-3(5a)^2}}{10}$$

Now we require:

$$14^2-3(5a)^2\ge0$$

which implies:

$$a\in\{-1,0,1\}$$

Case 1: $$a=-1$$

$$b=\frac{19\pm11}{10}$$

The integral value is $$b=3$$

Hence, $(a,b)=(-1,3)$ is a solution.

Case 2: $$a=0$$

$$b=\frac{14\pm14}{10}$$

The integral value is $$b=0$$

Hence, $(a,b)=(0,0)$ is a solution.

Case 3: $$a=1$$

$$b=\frac{9\pm11}{10}$$

The integral value is $$b=2$$

Hence, $(a,b)=(1,2)$ is a solution.

thanks ,your answer is quite right(Yes)

 

[Albert1]

my solution:

Spoiler

let x=2a+b,and y=a+2b
$ \therefore a=\dfrac {2x-y}{3},\,\,b=\dfrac {2y-x}{3}$
$5(a^2+ab+b^2)=7a+14b,\,\,becomes:$
$5(x^2-xy+y^2)=21y$
and we get :$y=5---(1),\,\, x^2-xy+y^2=21---(2)$
from (1)(2) (x,y)=(1,5)and (4,5)
and the corresponding (a,b)=(-1,3) and (1,2)
of course the third solution of (a,b)=(0,0)

 

[kaliprasad]

my solution:

Spoiler

let x=2a+b,and y=a+2b
$ \therefore a=\dfrac {2x-y}{3},\,\,y=\dfrac {2y-x}{3}$
$5(a^2+ab+b^2)=7a+14b,\,\,becomes:$
$5(x^2-xy+y^2)=21y$
and we get :$y=5---(1),\,\, x^2-xy+y^2=21---(2)$
from (1)(2) (x,y)=(1,5)and (4,5)
and the corresponding (a,b)=(-1,3) and (1,2)
of course the third solution of (a,b)=(0,0)

not correct as below

Spoiler

$5(x^2−xy+y^2)=21y$
and we get :y=5−−−(1),$x^2−xy+y^2=21$−−−(2)

is not strictly correctit could be

y=5k−−−(1),$x^2−xy+y^2=21k$−−−(2)

 

[Albert1]

not correct as below

Spoiler

$5(x^2−xy+y^2)=21y$
and we get :y=5−−−(1),$x^2−xy+y^2=21$−−−(2)

is not strictly correctit could be

y=5k−−−(1),$x^2−xy+y^2=21k$−−−(2)

$5(x^2−xy+y^2)=21y----(@)$
we get :$21y\geq 5xy$ $(AP\geq GP)$
$\therefore x\leq 4$ (x,y being integers)
if x=4 ,y=5k
from (@) :$16-20k+25k^2=21k$
$16+25k^2=41k$
k must =1

 

[kaliprasad]

$5(x^2−xy+y^2)=21y----(@)$
we get :$21y\geq 5xy$ $(AP\geq GP)$
$\therefore x\leq 4$ (x,y being integers)
if x=4 ,y=5k
from (@) :$16-20k+25k^2=21k$
$16+25k^2=41k$
k must =1

could you explain me how

we get :21y≥5xy (AM≥GM) AM and GM of what ?

I an sorry that I could not understand

 

[Albert1]

could you explain me how

we get :21y≥5xy (AM≥GM) AM and GM of what ?

I an sorry that I could not understand

$21y=5(x^2-xy+y^2)\geq 5(2xy-xy)=5xy$

$ for: \,\, x^2+y^2\geq 2xy \,\, (AM\geq GM) $