MHB What Are the Real Solutions for This Equation?

  • Thread starter Thread starter anemone
  • Start date Start date
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Solve for real solution(s) for $$x^2− x + 1 = (x^2+ x + 1)(x^2+ 2x + 4)$$.
 
Mathematics news on Phys.org
anemone said:
Solve for real solution(s) for $$x^2− x + 1 = (x^2+ x + 1)(x^2+ 2x + 4)$$.

My solution:

If we expand and collect like terms, we have:

$$x^4+3x^3+6x^2+7x+3=0$$

Let's assume the LHS factors in the form:

$$x^4+3x^3+6x^2+7x+3=(x^2+ax+1)(x^2+bx+3)=x^4+(a+b)x^3+(ab+4)x^2+(3a+b)x+3$$

Equating coefficients selected to result in a linear 2X2 system, we have:

$$a+b=3$$

$$3a+b=7$$

From this we find:

$$(a,b)=(2,1)$$

Hence:

$$x^4+3x^3+6x^2+7x+3=(x^2+2x+1)(x^2+x+3)=(x+1)^2(x^2+x+3)=0$$

The discriminant of the second factor is negative, thus the only real solution is:

$$x=-1$$
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Replies
2
Views
1K
Replies
7
Views
1K
Replies
6
Views
1K
Replies
6
Views
1K
Back
Top