Solve x^3=3375

$$x ^ { 3 } = 3375$$

Solve for x (complex solution)

$$x=\frac{-15\sqrt{3}i-15}{2}\approx -7.5-12.990381057i$$

$$x=15$$

$$x=\frac{-15+15\sqrt{3}i}{2}\approx -7.5+12.990381057i$$

Solution Steps

Subtract 3375 from both sides.

$$x^{3}-3375=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -3375 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±3375,±1125,±675,±375,±225,±135,±125,±75,±45,±27,±25,±15,±9,±5,±3,±1$$

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.

$$x=15$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-3375 by x-15 to get x^{2}+15x+225. Solve the equation where the result equals to 0.

$$x^{2}+15x+225=0$$

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, 15 for b, and 225 for c in the quadratic formula.

$$x=\frac{-15±\sqrt{15^{2}-4\times 1\times 225}}{2}$$

Do the calculations.

$$x=\frac{-15±\sqrt{-675}}{2}$$

Solve the equation x^{2}+15x+225=0 when ± is plus and when ± is minus.

$$x=\frac{-15i\sqrt{3}-15}{2} x=\frac{-15+15i\sqrt{3}}{2}$$

List all found solutions.

$$x=15 x=\frac{-15i\sqrt{3}-15}{2} x=\frac{-15+15i\sqrt{3}}{2}$$

Solve for x

$$x=15$$

Solution Steps

Subtract 3375 from both sides.

$$x^{3}-3375=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -3375 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±3375,±1125,±675,±375,±225,±135,±125,±75,±45,±27,±25,±15,±9,±5,±3,±1$$

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.

$$x=15$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-3375 by x-15 to get x^{2}+15x+225. Solve the equation where the result equals to 0.

$$x^{2}+15x+225=0$$

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, 15 for b, and 225 for c in the quadratic formula.

$$x=\frac{-15±\sqrt{15^{2}-4\times 1\times 225}}{2}$$

Do the calculations.

$$x=\frac{-15±\sqrt{-675}}{2}$$

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.

$$x=15$$