Solve x^3+3x^2-x-6=0

$${ x }^{ 3 } +3 { x }^{ 2 } -x-6 = 0$$

Solve for x

$$x = \frac{\sqrt{13} - 1}{2} \approx 1.302775638$$

$$x=\frac{-\sqrt{13}-1}{2}\approx -2.302775638$$

$$x=-2$$

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 -6 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±6,±3,±2,±1$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+3x^{2}-x-6 by x+2 to get x^{2}+x-3. Solve the equation where the result equals to 0.

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

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

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

Do the calculations.

$$x=\frac{-1±\sqrt{13}}{2}$$

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

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

List all found solutions.

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