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### When is the inequality x^4 < 8x^2 + 9 true?

If we consider this like a normal quadratic problem, this becomes easy

x4 < 8x2 + 9

x4 - 8x2 - 9 < 0

(x2-9)(x2+1) < 0

This means there are roots of this expression at x2 = 9 and x2 = -1

Since for all reals, x2 > 0, we know the two roots of this expression are x=+-3

Now, since x4 - 8x2 - 9 is a quartic (ie, it has an x4 expression), we know that given any sufficiently positive or negative x, the quartic will be positive (ie, if x is 10000, or -10000)

Therefore, we know for this to be true, -3

2 years ago

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