Solve x^3-37x+6=0

$$x ^ { 3 } - 37 x + 6 = 0$$

Solve for x (complex solution)

$$x=-\left(\sqrt{10}+3\right)\approx -6.16227766$$

$$x=6$$

$$x=\sqrt{10}-3\approx 0.16227766$$

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.

$$x=6$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-37x+6 by x-6 to get x^{2}+6x-1. Solve the equation where the result equals to 0.

$$x^{2}+6x-1=0$$

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

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

Do the calculations.

$$x=\frac{-6±2\sqrt{10}}{2}$$

Solve the equation x^{2}+6x-1=0 when ± is plus and when ± is minus.

$$x=-\sqrt{10}-3 x=\sqrt{10}-3$$

List all found solutions.

$$x=6 x=-\sqrt{10}-3 x=\sqrt{10}-3$$

Solve for x

$$x=-\sqrt{10}-3\approx -6.16227766$$

$$x=6$$

$$x=\sqrt{10}-3\approx 0.16227766$$

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.

$$x=6$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-37x+6 by x-6 to get x^{2}+6x-1. Solve the equation where the result equals to 0.

$$x^{2}+6x-1=0$$

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

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

Do the calculations.

$$x=\frac{-6±2\sqrt{10}}{2}$$

Solve the equation x^{2}+6x-1=0 when ± is plus and when ± is minus.

$$x=-\sqrt{10}-3 x=\sqrt{10}-3$$

List all found solutions.

$$x=6 x=-\sqrt{10}-3 x=\sqrt{10}-3$$