Solve n(n^2-1)/6=10

$$\frac { n ( n ^ { 2 } - 1 ) } { 6 } = 10$$

Solve for n (complex solution)

$$n=-\sqrt{11}i-2\approx -2-3.31662479i$$

$$n=4$$

$$n=-2+\sqrt{11}i\approx -2+3.31662479i$$

Solution Steps

Multiply both sides by 6.

$$n\left(n^{2}-1\right)=10\times 6$$

Use the distributive property to multiply n by n^{2}-1.

$$n^{3}-n=10\times 6$$

Multiply 10 and 6 to get 60.

$$n^{3}-n=60$$

Subtract 60 from both sides.

$$n^{3}-n-60=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -60 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±60,±30,±20,±15,±12,±10,±6,±5,±4,±3,±2,±1$$

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.

$$n=4$$

By Factor theorem, n-k is a factor of the polynomial for each root k. Divide n^{3}-n-60 by n-4 to get n^{2}+4n+15. Solve the equation where the result equals to 0.

$$n^{2}+4n+15=0$$

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, 4 for b, and 15 for c in the quadratic formula.

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

Do the calculations.

$$n=\frac{-4±\sqrt{-44}}{2}$$

Solve the equation n^{2}+4n+15=0 when ± is plus and when ± is minus.

$$n=-\sqrt{11}i-2 n=-2+\sqrt{11}i$$

List all found solutions.

$$n=4 n=-\sqrt{11}i-2 n=-2+\sqrt{11}i$$

Solve for n

$$n=4$$

Solution Steps

Multiply both sides by 6.

$$n\left(n^{2}-1\right)=10\times 6$$

Use the distributive property to multiply n by n^{2}-1.

$$n^{3}-n=10\times 6$$

Multiply 10 and 6 to get 60.

$$n^{3}-n=60$$

Subtract 60 from both sides.

$$n^{3}-n-60=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -60 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±60,±30,±20,±15,±12,±10,±6,±5,±4,±3,±2,±1$$

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.

$$n=4$$

By Factor theorem, n-k is a factor of the polynomial for each root k. Divide n^{3}-n-60 by n-4 to get n^{2}+4n+15. Solve the equation where the result equals to 0.

$$n^{2}+4n+15=0$$

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, 4 for b, and 15 for c in the quadratic formula.

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

Do the calculations.

$$n=\frac{-4±\sqrt{-44}}{2}$$

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

$$n\in \emptyset$$

List all found solutions.

$$n=4$$