/sci/ - Science & Math

[Anonymous No.1983171]

Guys, how do I solve this.
<span class="math">a+b+c = 18[/spoiler]
<span class="math">a^2+b^2+c^2 = 756[/spoiler]
<span class="math">a^2=bc[/spoiler]

I couldn't get past:
<span class="math">a^2+b^2+c^2 = 42(a+b+c)[/spoiler]
Just to let you know, one can easily find the answers using wolfram, but I want to figure out how to do it without a computer.

[Anonymous No.1983182]

Bumping. Keep in mind I already got the answers.

[Anonymous No.1983190]

Square expression one, sub shit in, rearrange for a.

[Anonymous No.1983191]

Bump

[Anonymous No.1983194]

<div class="math">b + c = 18 - a</div>
<div class="math">(b + c)^2 = (18 - a)^2</div>
and
<div class="math">(b + c)^2 = b^2 + c^2 + 2bc = 756 + a^2</div>
you can find a now
<div class="math">18^2 - 36a = 756</div>
<div class="math">a = (756 - 18^2) / 36</div>
then you know b+c and bc and can find b and c

[Anonymous No.1983196]

Differential equations

[Anonymous No.1983205]

>>1983190

Squaring that would give me. a^2 + 2ab +2ac + b^2 + 2bc + c^2 = 324

[Anonymous No.1983207]

a^2=bc iff b = c = a

[Anonymous No.1983211]

>>1983194

>18^2 - 36a = 756

How?

[Anonymous No.1983213]

>>1983171
Greatest common divisors. Number theory up that motherfucker.

[Anonymous No.1983214]

>>1983211
>mfw you can't read the 2 lines above

[Anonymous No.1983216]

3 equations 3 variables .... express first one through equation of your joice, substitute expressed variable in one of the equations not used until now, express one of the two remaining variables, substitute in last equation now you have a value for one of the variables ....

[Anonymous No.1983218]

a + b + c = 18
a^2 + b^2 + c^2 = 756
a^2 = bc

(a + b + c)^2 = 18^2
a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = 324
2ab + 2ac + 2bc = 324
ab + ac + bc = 162
ab + ac + a^2 = 162
a * (a + b + c) = 162
a * (18) = 162
a = 9

9 + b + c = 18
81 + b^2 + c^2 = 756
81 = bc

b + c = 9
b^2 + c^2 = 675
81 = bc

81/b = c
b + 81/b = 9
b^2 + 81 = 9b
b^2 - 9b + 81 = 0
b = (9 +/- sqrt(81 - 324)) / 2
b = (9 +/- sqrt(-243)) / 2
b = (9 +/- 9i * sqrt(3)) / 2
b = (9/2) * (1 +/- i * sqrt(3))

b + c = 9
(9/2) * (1 +/- i * sqrt(3)) + c = 9
c = 9 - (9/2) * (1 +/- i * sqrt(3))
c = (9/2) * (1 +/- i * sqrt(3))

a = 9
b = (9/2) * (1 + i * sqrt(3))
c = (9/2) * (1 - i * sqrt(3))

[Anonymous No.1983228]

(a+b+c)^2 = 18^2

(a+b+c)^2 - (a^2 + b^2 +c^2)= (ab+ac+bc) =(ab +ac +a^2)
=a(b+c+a) = a * 18 = 18^2 - 756 = -24 * 18

a= -24 etc

[Anonymous No.1983226]

>>1983194
This is the answer. I confirm it.

[Anonymous No.1983236]

>>1983218
>a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = 324
>2ab + 2ac + 2bc = 324
Wait, what?

Also, your answers are wrong. Only natural numbers.

[Anonymous No.1983245]

>>1983236
well there sparky, thats a different problem then, isnt it

[Anonymous No.1983249]

>>1983194

Someone explain the second line?

[Anonymous No.1983252]

And the third too, please. Why do those two things equal each other? From which expression did it come?

[Anonymous No.1983257]

>>1983252
>>1983249
directly from the statement
hope you are trolling

captcha: embulan threadbare

[Anonymous No.1983260]

>>1983257

No, not really. When I do what he did, I get:
b^2 + c^2 + 2a^2 = 18^2 + a^2
b^2 + c^2 + a^2 = 18^2

Which too me means that 756 = 18^2; which is false. I guess I fail here.

[Anonymous No.1983267]

help appreciagted. I still don't get it.

[Anonymous No.1983268]

>>1983267

[aether No.1983299]

{herpa derpa}

[Anonymous No.1983315]

>mfw /sci/ cannot do basic linear algebra

derp derp herp

[aether No.1983325]

>>1983315
>mfw you dunno what mfw means

[Anonymous No.1983332]

(a+b+c)^2=a^2+b^2+c^2+2(ab+ac+bc)=18^2
(ab+ac+bc)=(18^2-756)/2=-216
(ab+ac+bc)=(ab+ac+a^2)=a(a+b+c)=-216
a=-216/18=-12

now:

b+c=30
bc=144

b=(30-c)
(30-c)c=144

c^2-30c+144=0

c=(30 +/- sqrt(900-4*144))/2

c=24 or c=6
b=6 or b=24

[Anonymous No.1983368]

Just generate the groebner basis for the ideal
<span class="math"><a + b + c - 18, a^2 + b^2 + c^2 - 756, a^2-bc> \subseteq \mathbb Q[a,b,c][/spoiler]


I get:

<span class="math"> I = < a + 12, b + c - 30, c^2 - 30*c + 144 >[/spoiler]

Which implies two solutions <span class="math"> a = -12, b = 24, c = 6[/spoiler] and <span class="math"> a =-12, b = 6, c = 24[/spoiler]

Hope this helps!