Challenge Quadratic equation that has two roots

anemone

Paris la ville de l'amour
Staff member
Administrator
Moderator
Math Helper
Jan 28, 2018
181
219
43
Find all integer valuse of \(\displaystyle k\) such that the quadratic expression \(\displaystyle (x+k)(x+1991)+1\) can be factored as a product of \(\displaystyle (x+p)(x+q)\), where \(\displaystyle p\) and \(\displaystyle q\) are integers.
 
  • Like
Reactions: MarkFL

MarkFL

La Villa Strangiato
Staff member
Administrator
Moderator
Math Helper
Jan 25, 2018
3,487
4,264
113
St. Augustine
If we expand and equate the two expressions, we obtain:

\(\displaystyle x^2+(k+1991)x+1991k+1=x^2+(p+q)x+pq\)

Equating coefficients, we get:

\(\displaystyle k+1991=p+q\)

\(\displaystyle 1991k+1=pq\)

Now, the second equation implies:

\(\displaystyle q=\frac{1991k+1}{p}\)

Substituting into the first equation, we obtain:

\(\displaystyle k+1991=p+\frac{1991k+1}{p}\)

Multiply by \(p\) and arrange in standard form:

\(\displaystyle p^2-(k+1991)p+1991k+1=0\)

One condition we require for the roots of this quadratic to be an integer is for \(k\) to be odd, so let:

\(\displaystyle k=2r+1\) where \(r\in\mathbb{Z}\)

And we have:

\(\displaystyle p^2-(2r+1+1991)p+1991(2r+1)+1=0\)

Or:

\(\displaystyle p^2-2(r+996)p+2(1991r+996)=0\)

Now, we also require the discriminant to be a perfect square:

\(\displaystyle (-2(r+996))^2-4(1)2(1991r+996)=2^2\left((r+996)^2-2(1991r+996)\right)=2^2(r-994)(r-996)\)

Let's let:

\(\displaystyle s=r-996\)

And look at the following product as a perfect square:

\(\displaystyle s(s+2)=n^2\)

\(\displaystyle s^2+2s-n^2=0\)

The discriminant here is:

\(\displaystyle 4+4n^2=4(n^2+1)\)

The only two perfect squares that differ by 1 are 0 and 1, and so we must have \(n=0\). This implies:

\(\displaystyle s=0\implies r=996\)

\(\displaystyle s=-2\implies r=994\)

Now, these together imply:

\(\displaystyle k\in\{1989,1993\}\)
 
  • Like
Reactions: anemone

anemone

Paris la ville de l'amour
Staff member
Administrator
Moderator
Math Helper
Jan 28, 2018
181
219
43
Very good Mark!(Cool) And thanks for participating in this challenge problem!(Heart)
 
  • Like
Reactions: MarkFL