kaliprasad said:
kaliprasad said:
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...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!
A system of equations is a set of equations that are related to each other and have common variables. The goal is to find values for the variables that satisfy all of the equations in the system.
The most common method for solving a system of equations is by substitution. This involves solving one equation for a variable and then substituting that value into the other equations. Another method is elimination, where you add or subtract equations to eliminate a variable. There are also advanced methods like Gaussian elimination and Cramer's rule.
Sure, let's say we have the system of equations a+b=5, ab+c+d=10, ad+bc=8, and cd=3. First, we can solve for a in the first equation to get a=5-b. Then, we can substitute that into the second equation to get (5-b)b+c+d=10. We can simplify this to -b^2+c+d-5b+10=0. Similarly, we can solve for d and substitute into the third equation to get (5-b)(c+b)=8. Finally, we can substitute these values into the fourth equation to get (5-b)(c+b)b=3. We can then solve for b, and use that value to find the other variables.
Yes, a system of equations can have one solution, no solutions, or infinitely many solutions. If the equations are inconsistent, meaning they contradict each other, there will be no solutions. If the equations are dependent, meaning one equation is a multiple of another, there will be infinitely many solutions. Otherwise, there will be one unique solution.
Solving systems of equations is used in many fields such as engineering, economics, physics, and chemistry. For example, in engineering, systems of equations can be used to determine the optimal design for a structure. In economics, systems of equations can be used to model supply and demand in a market. In physics, they can be used to predict the motion of objects. In chemistry, they can be used to analyze chemical reactions.