solve the following question: ||x^2-2x-8|+x^4-x^3+4x+8| =6

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Solution:

∣x2−2×x−8∣+x4−x3+4×x+8=6or∣x2−2×x−8∣+x4−x3+4×x+8=−6|x^2-2times x-8|+x^4-x^3+4times x+8=6 or |x^2-2times x-8|+x^4-x^3+4times x+8=-6∣x2−2×x−8∣+x4−x3+4×x+8=6or∣x2−2×x−8∣+x4−x3+4×x+8=−6

We construct graphs of the function where the polynomial obtained as a result of the transformation of the original equation is contained in the right-hand side. The equation will have solutions if there are intersection points of these graphs with the abscissa axis.

∣x2−2×x−8∣+x4−x3+4×x+8=6|x^2-2times x-8|+x^4-x^3+4times x+8=6∣x2−2×x−8∣+x4−x3+4×x+8=6

if x2−2×x−8≥0x^2-2times x-8ge0x2−2×x−8≥0

The resulting equation has no roots.

x4−x3+x2+2×x−6=0x^4-x^3+x^2+2times x-6=0x4−x3+x2+2×x−6=0

illustrate this on the graphs

minimum 18 if

       x leq-2    Does not cross the abscissa axis.

minimum 110 Does not cross the abscissa axis.

x2−2×x−8≤0x^2-2times x-8leq0x2−2×x−8≤0

we get the equation

x4−x3+6×x+10=0,−2<x<4x^4-x^3+6times x+10=0, -2<x<4x4−x3+6×x+10=0,−2<x<4

minimum 6ю Does not cross the abscissa axis.

∣x2−2×x−8∣+x4−x3+4×x+8=−6|x^2-2times x-8|+x^4-x^3+4times x+8=-6∣x2−2×x−8∣+x4−x3+4×x+8=−6

if x2−2×x−8≥0x^2-2times x-8ge0x2−2×<span class=