MHB Finding Integer Roots of $h+k=2016$

  • Thread starter Thread starter Albert1
  • Start date Start date
  • Tags Tags
    Integer Roots
AI Thread Summary
To find integer roots of the equation \(x^2 + hx + k = 0\) given \(h + k = 2016\), it is essential to express \(k\) in terms of \(h\) as \(k = 2016 - h\). The roots \(\alpha\) and \(\beta\) can be derived using Vieta's formulas, where \(\alpha + \beta = -h\) and \(\alpha \beta = k\). By substituting \(k\) into the product equation, the relationship becomes \(\alpha \beta = 2016 - h\). Solving these equations will yield possible integer values for \(h\), \(k\), \(\alpha\), and \(\beta\). The discussion emphasizes the need for integer solutions that satisfy both the sum and product conditions.
Albert1
Messages
1,221
Reaction score
0
given $h+k=2016$, and the two roots $\alpha \,\, and \,\, \beta $ of equation $x^2+hx+k=0$ are all integers , please find the value of:
$h,k,\alpha \,\, and \,\, \beta$
 
Last edited:
Mathematics news on Phys.org
Albert said:
given $h+k=2016$, and the two roots $\alpha \,\, and \,\, \beta $ of equation $x^2+hx+k=0$ are all integers , please find the value of:
$h,k,\alpha \,\, and \,\, \beta$

The roots are $\alpha,\beta$
$(x - \alpha) (x- \beta) = 0 $ or $x^2-(\alpha+ \beta) + \alpha\beta = 0$
comparing with given eqaution
$\alpha+ \beta= - h $ and $\alpha\beta= k$
from h + k = 2106 we get $\alpha\beta - (\alpha + \beta) = 2016 $
or $\alpha\beta - (\alpha + \beta) + 1 = 2017 $
or $(\alpha- 1)(\beta - 1) = 2017 $
as 2017 is a prime $\alpha = 2018$ and $\beta = 2$ and hence $h = - 2020,k = 4036$
or $\alpha = 2$ and $\beta = 2018$ and hence $h = - 2020,k = 4036$
 
$\alpha=2016,\,\beta=0,\,h=2016,\,k=0$ and any permutation thereof are also solutions.
Edit: sign error - the above is incorrect, sorry about that...:o
 
Last edited:
greg1313 said:
$\alpha=2016,\,\beta=0,\,h=2016,\,k=0$ and any permutation thereof are also solutions.

Thanks Greg

forgot the product to be (-1) and (-2017) which gives the above solution subject to the restriction that permutation of
$\alpha,\beta$ is possible but not any permutation
 
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...
Back
Top