| Sorry, theoneo,
seems your sleepy hands could not hold a pen any longer... I mean, keyboard.
If you don't mind, squeezing your narrative:
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If n > 0 and 9x^2 + kx + 36 = (3x+n)^2 for all values of x, what is the value of k - n?
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9x^2 + kx + 36 = 9x^2 + 6xn + n^2
kx + 36 = 6nx + n^2
Left and right sides of this equation are equations of straight lines for any set pair of values of k and n:
y1 = kx + 36
and
y2 = 6nx + n^2.
y1 = y2 for all values of x; that means y1 and y2 coincide:
k=6n and 36=n^2,
n=6 (since n>0),
k-n = 30. |