Solve x^4-4x^2-4x-1=0

$$x ^ { 4 } - 4 x ^ { 2 } - 4 x - 1 = 0$$

Solve for x

$$x=1-\sqrt{2}\approx -0.414213562$$

$$x=-1$$

$$x=\sqrt{2}+1\approx 2.414213562$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-4x^{2}-4x-1 by x+1 to get x^{3}-x^{2}-3x-1. Solve the equation where the result equals to 0.

$$x^{3}-x^{2}-3x-1=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-x^{2}-3x-1 by x+1 to get x^{2}-2x-1. Solve the equation where the result equals to 0.

$$x^{2}-2x-1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -2 for b, and -1 for c in the quadratic formula.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 1\left(-1\right)}}{2}$$

Do the calculations.

$$x=\frac{2±2\sqrt{2}}{2}$$

Solve the equation x^{2}-2x-1=0 when ± is plus and when ± is minus.

$$x=1-\sqrt{2} x=\sqrt{2}+1$$

List all found solutions.

$$x=-1 x=1-\sqrt{2} x=\sqrt{2}+1$$