Solve x^3-6x^2-2x+12=0

$${ x }^{ 3 } -6 { x }^{ 2 } -2x+12=0$$

Solve for x

$$x=-\sqrt{2}\approx -1.414213562$$

$$x=6$$

$$x=\sqrt{2}\approx 1.414213562$$

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

$$±12,±6,±4,±3,±2,±1$$

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.

$$x=6$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-6x^{2}-2x+12 by x-6 to get x^{2}-2. Solve the equation where the result equals to 0.

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

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, 0 for b, and -2 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 1\left(-2\right)}}{2}$$

Do the calculations.

$$x=\frac{0±2\sqrt{2}}{2}$$

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

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

List all found solutions.

$$x=6 x=-\sqrt{2} x=\sqrt{2}$$