25х^4-50х^3+14х^2+10х+1 <=0

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вопрос

To solve this inequality, we can factor the expression as follows:

25x^4 - 50x^3 + 14x^2 + 10x + 1 <= 0
=> x^4 - 2x^3 + 0.56x^2 + 0.4x + 0.04 <= 0
=> (x^2 - 1.4x + 0.2)(x^2 - 0.6x + 0.2) <= 0

Now we can find the roots of the quadratic equations x^2 - 1.4x + 0.2 = 0 and x^2 - 0.6x + 0.2 = 0 by using the quadratic formula:

For x^2 - 1.4x + 0.2 = 0:
x = (1.4 ± sqrt(1.4^2 - 410.2)) / 2
x = (1.4 ± sqrt(1.96 - 0.8)) / 2
x = (1.4 ± sqrt(1.16)) / 2
x = (1.4 ± 1.08) / 2

So, the roots are:
x = (1.4 + 1.08) / 2 = 1.74
x = (1.4 - 1.08) / 2 = 0.16

For x^2 - 0.6x + 0.2 = 0:
x = (0.6 ± sqrt(0.6^2 - 410.2)) / 2
x = (0.6 ± sqrt(0.36 - 0.8)) / 2
x = (0.6 ± sqrt(0.44)) / 2
x = (0.6 ± 0.66) / 2

So, the roots are:
x = (0.6 + 0.66) / 2 = 0.63
x = (0.6 - 0.66) / 2 ≈ -0.03

Now we know that the solutions to the inequality lie between the roots of the quadratic equations. So the solution to the inequality is:
-0.03 <= x <= 0.16 or 0.63 <= x <= 1.74.