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perfect square
- Thread starter suvadip
- Start date
Hello, suvadip!
A quadratic is a square if its discriminant, \(\displaystyle b^2\!-\!4ac\), is zero.
\(\displaystyle D \;=\;[4(y+1)]^2 - 4(y+16)\left(\frac{y^2-128}{4}\right)\)
. . .\(\displaystyle =\;16(y^2+2y+1) - (y+16)(y^2-128)\)
. . .\(\displaystyle =\;16y^2 + 32y + 16 - (y^3 + 16y^2 - 128y - 2048)\)
. . .\(\displaystyle =\;16y^2 + 32y + 16 - y^3 - 16y^2 + 128y + 2048\)
. . .\(\displaystyle =\;-y^3 + 160y + 2064\)
\(\displaystyle \text{If }y^3 - 160y - 2064 \:=\:0,\, \text{ the only rational roots are:}\) \(\displaystyle 1,2,4,6,8,16,18,24,\,\hdots\)
\(\displaystyle \text{If }y = 16,\,\text{ we get: }\,-528\)
\(\displaystyle \text{If }y = 18,\,\text{ we get: }\,+888\)
There is an irrational root between 16 and 18.
. . Good luck!
\(\displaystyle \text{For real value of }y\text{ is }\,(y+16)x^2 - 4(y+1)x - \dfrac{y^2-128}{4}\,\text{ a perfect square?}\)
A quadratic is a square if its discriminant, \(\displaystyle b^2\!-\!4ac\), is zero.
\(\displaystyle D \;=\;[4(y+1)]^2 - 4(y+16)\left(\frac{y^2-128}{4}\right)\)
. . .\(\displaystyle =\;16(y^2+2y+1) - (y+16)(y^2-128)\)
. . .\(\displaystyle =\;16y^2 + 32y + 16 - (y^3 + 16y^2 - 128y - 2048)\)
. . .\(\displaystyle =\;16y^2 + 32y + 16 - y^3 - 16y^2 + 128y + 2048\)
. . .\(\displaystyle =\;-y^3 + 160y + 2064\)
\(\displaystyle \text{If }y^3 - 160y - 2064 \:=\:0,\, \text{ the only rational roots are:}\) \(\displaystyle 1,2,4,6,8,16,18,24,\,\hdots\)
\(\displaystyle \text{If }y = 16,\,\text{ we get: }\,-528\)
\(\displaystyle \text{If }y = 18,\,\text{ we get: }\,+888\)
There is an irrational root between 16 and 18.
. . Good luck!
Perhaps I was unable to make you understand. I need only the real value of y for which the given expression will be a perfect square. x should be present in the expression. I want to write the expression as (ax-b)2 . Actually I want to solve x4-8x3+4x+32=0 by Ferrari method. Thats why I need such y.
Huh? Is that the FULL original problem?
For now, the expression is a perfect square in infinite cases; like:
x=0, y=12 : 4
x=0, y=18 :49
x=1, y=36 : 196
x=2, y=6 : 9