MHB Solve System of Equations: a+b, ab+c+d, ad+bc, cd

[kaliprasad]

solve the following system of equations in real a,b,c,d

$a+ b =8 $

$ab + c +d = 23 $

$ad + bc = 28$

$cd = 12$

 

[kaliprasad]

No answer yet

hint

Spoiler

form a quartic equation

 

[anemone]

solve the following system of equations in real a,b,c,d

$a+ b =8 $

$ab + c +d = 23 $

$ad + bc = 28$

$cd = 12$

No answer yet

hint

Spoiler

form a quartic equation

My solution:

Spoiler

If we form a quartic equation as the product of two quadratic equations as follow, we have:

$\begin{align*}(x^2+ax+c)(x^2+bx+d)&=x^4+(a+b)x^3+(ab+c+d)x^2+(ad+bc)x+cd\\&=x^4+8x^3+23x^2+28x+12\\&\overset{I}{=}((x+1)(x+2))((x+2)(x+3))=(x^2+3x+2)(x^2+5x+6)\\&\overset{II}{=}((x+1)(x+3))((x+2)(x+2))=(x^2+4x+3)(x^2+4x+4)\\&\overset{III}{=}((x+2)(x+3))((x+1)(x+2))=(x^2+5x+6)(x^2+3x+2)\\&\overset{IV}{=}((x+2)(x+2))((x+1)(x+3))=(x^2+4x+4)(x^2+4x+3)\end{align*}$

Hence, $(a,\,b,\,c,\,d)=(3,\,5,\,2,\,6),\,(4,\,4,\,3,\,4),\,(5,\,3,\,6,\,2),\,(4,\,4,\,4,\,3)$.

But...I'm not proud of myself for this solution because...

Spoiler

Without the hint, I could have never solved this great challenge!Thank you so much, kaliprasad for the hint and also for sharing with us of this great challenge! I could tell I've fallen in love with this problem at first sight!Hahaha...

 

[kaliprasad]

My solution:

Spoiler

If we form a quartic equation as the product of two quadratic equation as follow, we have:

$\begin{align*}(x^2+ax+c)(x^2+bx+d)&=x^4+(a+b)x^3+(ab+c+d)x^2+(ad+bc)x+cd\\&=x^4+8x^3+23x^2+28x+12\\&\overset{I}{=}((x+1)(x+2))((x+2)(x+3))=(x^2+3x+2)(x^2+5x+6)\\&\overset{II}{=}((x+1)(x+3))((x+2)(x+2))=(x^2+4x+3)(x^2+4x+4)\\&\overset{III}{=}((x+2)(x+3))((x+1)(x+2))=(x^2+5x+6)(x^2+3x+2)\\&\overset{IV}{=}((x+2)(x+2))((x+1)(x+3))=(x^2+4x+4)(x^2+4x+3)\end{align*}$

Hence, $(a,\,b,\,c,\,d)=(3,\,5,\,2,\,6),\,(4,\,4,\,3,\,4),\,(5,\,3,\,6,\,2),\,(4,\,4,\,4,\,3)$.

But...I'm not proud of myself for this solution because...

Spoiler

Without the hint, I could have never solved this great challenge!Thank you so much, kaliprasad for the hint and also for sharing with us of this great challenge! I could tell I've fallen in love with this problem at first sight!Hahaha...

Nothing more to write to make it right(pun intended)

Hence it is closed