Solve x^3-21x-90=0

$${ x }^{ 3 } -21x-90 = 0$$

Solve for x (complex solution)

$$x=-\sqrt{6}i-3\approx -3-2.449489743i$$

$$x=6$$

$$x=-3+\sqrt{6}i\approx -3+2.449489743i$$

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 -90 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±90,±45,±30,±18,±15,±10,±9,±6,±5,±3,±2,±1$$

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}-21x-90 by x-6 to get x^{2}+6x+15. Solve the equation where the result equals to 0.

$$x^{2}+6x+15=0$$

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 15 for c in the quadratic formula.

$$x=\frac{-6±\sqrt{6^{2}-4\times 1\times 15}}{2}$$

Do the calculations.

$$x=\frac{-6±\sqrt{-24}}{2}$$

Solve the equation x^{2}+6x+15=0 when ± is plus and when ± is minus.

$$x=-\sqrt{6}i-3 x=-3+\sqrt{6}i$$

List all found solutions.

$$x=6 x=-\sqrt{6}i-3 x=-3+\sqrt{6}i$$

Solve for x

$$x=6$$

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 -90 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±90,±45,±30,±18,±15,±10,±9,±6,±5,±3,±2,±1$$

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}-21x-90 by x-6 to get x^{2}+6x+15. Solve the equation where the result equals to 0.

$$x^{2}+6x+15=0$$

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 15 for c in the quadratic formula.

$$x=\frac{-6±\sqrt{6^{2}-4\times 1\times 15}}{2}$$

Do the calculations.

$$x=\frac{-6±\sqrt{-24}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$x\in \emptyset$$

List all found solutions.

$$x=6$$