Solve x^3-10x-12=0

$$x ^ { 3 } - 10 x - 12 = 0$$

Solve for x

$$x=\sqrt{7}+1\approx 3.645751311$$

$$x=1-\sqrt{7}\approx -1.645751311$$

$$x=-2$$

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 -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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-10x-12 by x+2 to get x^{2}-2x-6. Solve the equation where the result equals to 0.

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

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 -6 for c in the quadratic formula.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 1\left(-6\right)}}{2}$$

Do the calculations.

$$x=\frac{2±2\sqrt{7}}{2}$$

Solve the equation x^{2}-2x-6=0 when ± is plus and when ± is minus.

$$x=1-\sqrt{7} x=\sqrt{7}+1$$

List all found solutions.

$$x=-2 x=1-\sqrt{7} x=\sqrt{7}+1$$