Solve x^3=1728

$$x ^ { 3 } = 1728$$

Solve for x (complex solution)

$$x=-6\sqrt{3}i-6\approx -6-10.392304845i$$

$$x=12$$

$$x=-6+6\sqrt{3}i\approx -6+10.392304845i$$

Solution Steps

Subtract 1728 from both sides.

$$x^{3}-1728=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1728 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±1728,±864,±576,±432,±288,±216,±192,±144,±108,±96,±72,±64,±54,±48,±36,±32,±27,±24,±18,±16,±12,±9,±8,±6,±4,±3,±2,±1$$

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.

$$x=12$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-1728 by x-12 to get x^{2}+12x+144. Solve the equation where the result equals to 0.

$$x^{2}+12x+144=0$$

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, 12 for b, and 144 for c in the quadratic formula.

$$x=\frac{-12±\sqrt{12^{2}-4\times 1\times 144}}{2}$$

Do the calculations.

$$x=\frac{-12±\sqrt{-432}}{2}$$

Solve the equation x^{2}+12x+144=0 when ± is plus and when ± is minus.

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

List all found solutions.

$$x=12 x=-6i\sqrt{3}-6 x=-6+6i\sqrt{3}$$

Solve for x

$$x=12$$

Solution Steps

Subtract 1728 from both sides.

$$x^{3}-1728=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1728 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±1728,±864,±576,±432,±288,±216,±192,±144,±108,±96,±72,±64,±54,±48,±36,±32,±27,±24,±18,±16,±12,±9,±8,±6,±4,±3,±2,±1$$

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.

$$x=12$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-1728 by x-12 to get x^{2}+12x+144. Solve the equation where the result equals to 0.

$$x^{2}+12x+144=0$$

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, 12 for b, and 144 for c in the quadratic formula.

$$x=\frac{-12±\sqrt{12^{2}-4\times 1\times 144}}{2}$$

Do the calculations.

$$x=\frac{-12±\sqrt{-432}}{2}$$

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.

$$x=12$$