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

Click For Summary
The discussion revolves around solving a system of equations involving four variables: a, b, c, and d. The equations provided are a + b = 8, ab + c + d = 23, ad + bc = 28, and cd = 12. Despite attempts to find a solution, no definitive answer has been reached, and participants express a lack of confidence in their solutions. The conversation indicates a struggle with the complexity of the equations, leading to a sense of closure without resolution. The thread highlights the challenges of solving nonlinear systems of equations.
kaliprasad
Gold Member
MHB
Messages
1,333
Reaction score
0
solve the following system of equations in real a,b,c,d

$a+ b =8 $

$ab + c +d = 23 $

$ad + bc = 28$

$cd = 12$
 
Mathematics news on Phys.org
No answer yet

hint

form a quartic equation
 
kaliprasad said:
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 said:
No answer yet

hint

form a quartic equation

My solution:

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...

Without the hint, I could have never solved this great challenge!:oThank 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...
 
Last edited:
anemone said:
My solution:

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...

Without the hint, I could have never solved this great challenge!:oThank 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
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

MHB Simplify a+b+c+d+2(ab+ac+ad+bc+bd+cd)+4(abc+abd+acd+bcd)+8(abcd)
  • · Replies 3 ·
Replies
3
Views
3K
MHB Evaluate Quotient: $ab+cd \over ad+bc$
  • · Replies 7 ·
Replies
7
Views
2K
MHB Prove Inequality: $\sqrt{ab}+\sqrt{cd}\le \sqrt{(a+d)(b+c)}$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Finding the Coordinates of Point D
  • · Replies 1 ·
Replies
1
Views
1K
MHB A,B,C,D cocyclic prove AC^2=AB×AD+BC^2
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Does this make a cubic out of a quadratic system?
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Solving a system of equations with 5 unknowns
  • · Replies 10 ·
Replies
10
Views
1K
MHB Inequality involving a, b, c and d
  • · Replies 1 ·
Replies
1
Views
900
MHB No Real Solutions to System of Equations
  • · Replies 3 ·
Replies
3
Views
2K
MHB Can you solve this system of equations with four variables?
  • · Replies 1 ·
Replies
1
Views
1K