Solve x^3+2x^2-4x+1=0

$${ x }^{ 3 } +2 { x }^{ 2 } -4x+1=0$$

Solve for x

$$x=\frac{-\sqrt{13}-3}{2}\approx -3.302775638$$

$$x=1$$

$$x=\frac{\sqrt{13}-3}{2}\approx 0.302775638$$

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^{3}+2x^{2}-4x+1 by x-1 to get x^{2}+3x-1. Solve the equation where the result equals to 0.

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

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, 3 for b, and -1 for c in the quadratic formula.

$$x=\frac{-3±\sqrt{3^{2}-4\times 1\left(-1\right)}}{2}$$

Do the calculations.

$$x=\frac{-3±\sqrt{13}}{2}$$

Solve the equation x^{2}+3x-1=0 when ± is plus and when ± is minus.

$$x=\frac{-\sqrt{13}-3}{2} x=\frac{\sqrt{13}-3}{2}$$

List all found solutions.

$$x=1 x=\frac{-\sqrt{13}-3}{2} x=\frac{\sqrt{13}-3}{2}$$