Question

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

Answer 1

Solution:

$|x^2-2\times x-8|+x^4-x^3+4\times x+8=6 or |x^2-2\times x-8|+x^4-x^3+4\times x+8=-6$

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.

$|x^2-2\times x-8|+x^4-x^3+4\times x+8=6$

if $x^2-2\times x-8\ge0$

The resulting equation has no roots.

$x^4-x^3+x^2+2\times x-6=0$

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.

$x^2-2\times x-8\leq0$

we get the equation

$x^4-x^3+6\times x+10=0, -2<x<4$

minimum 6ю Does not cross the abscissa axis.

$|x^2-2\times x-8|+x^4-x^3+4\times x+8=-6$

if $x^2-2\times x-8\ge0$ we will have eqation

$x^4-x^3+x^2+2\times x+6=0$

minimum 30 if   /Does not cross the abscissa axis.

minimum 222 if $x \ge 4$

if $x^2-2\times x-8\leq0$ . we have eqation

$x^4-x^3-x^2+6\times x+22=0, -2<x<4$

minimum 17.

Since no intersection points of the constructed graphs with the abscissa were obtained, the equation has no roots.

Answer: The equation has no roots.