How to find m^2 + 4n^2 =? from m^3 - 12mn^2 = 40 4n^3-3m^2n=10

m^3 - 12mn^2 = 40
4n^3-3m^2n=10

m^2 + 4n^2 =?

2 Answers
Jul 21, 2018

m^2+4n^2=10 root(3)(2)

Explanation:

I assume m and n are real
{(m^3-12mn^2=40), (4n^3-3m^2n=10):}

Factor out what we can:
{(m(m^2-12n^2)=40), (n(4n^2-3m^2)=10):}

I can see that n^2 appears 2 times with coefficients 4 and 12, so I could try to substitute k=2n or k^2=4n^2:
{(m(m^2-3*4n^2)=40), (n(4n^2-3m^2)=10):}

{(m(m^2-3color(red)(k^2))=40), (n(color(red)(k^2)-3m^2)=10):}

and double the second equation just for taste
{(m(m^2-3k^2)=40), (color(red)(k)(k^2-3m^2)=20):}

Distribute
{(m^3-3mk^2=40), (k^3-3m^2k=20):}

It looks very symmetrical now and closely related to cube of a sum. The problem is that adding or subtracting will never give perfect cube, besause signs wont match.

Maybe complex numbers can help?

Multiply second equation by i

{(m^3-3mk^2=40), (ik^3-3im^2k=20i):}

Adding gives
{:(m^3,-3im^2k,-3mk^2,+ik^3,=40+20i), (m^3,+3m^2(-i)k,+3m(-1)k^2,+(i)k^3,=40+20i), (m^3,+3m^2(-i)k,+3m(-i)^2k^2,+(-i)^3k^3,=40+20i):}

(m-ik)^3=40+20i=20(2+i)

Subtracting gives
{:(m^3,+3im^2k,-3mk^2,-ik^3,=40-20i), (m^3,+3m^2(i)k,+3m(-1)k^2,+(-i)k^3,=40-20i), (m^3,+3m^2(i)k,+3m(i)^2k^2,+(i)^3k^3,=40-20i):}

(m+ik)^3=40-20i=20(2-i)

Finally we have
{((m-ik)^3=20(2+i)), ((m+ik)^3=20(2-i)):}

Multiply them together using difference of squares
((m-ik)(m+ik))^3=400(2+i)(2-i)
(m^2+k^2)^3=400(4+1), where k=2n
(m^2+4n^2)^3=2000=2*10^3

Take cube root for the answer
m^2+4n^2=10 root(3)(2)

Jul 21, 2018

m^2+4n^2=10 root(3)(2)

Explanation:

It is a solution without using complex numbers. This solution is shorter and more direct, but it's less clear why it works.

Same as before until this point:
{(m^3-3mk^2=40), (k^3-3m^2k=20):}

Square both equations
{((m^3)^2-2*m^3*3mk^2+(3mk^2)^2=1600), ((k^3)^2-2*k^3*3m^2k+(3m^2k)^2=400):}

Simplify
{(m^6-6m^4k^2+9m^2k^4=1600), (k^6-6m^2k^4+9m^4k^2=400):}

Add them together
m^6+3m^4k^2+3m^2k^4+k^6=2000

Use cube of sum
(m^2+k^2)^3=2000, where k=2n
(m^2+4n^2)^3=2000=2*10^3

Take cube root for the answer
m^2+4n^2=10 root(3)(2)