Solve x^3+3x^2+3x=-28

$$x ^ { 3 } + 3 x ^ { 2 } + 3 x = - 28$$

Solve for x (complex solution)

$$x=\frac{-3\sqrt{3}i+1}{2}\approx 0.5-2.598076211i$$

$$x=-4$$

$$x=\frac{1+3\sqrt{3}i}{2}\approx 0.5+2.598076211i$$

Solution Steps

Add 28 to both sides.

$$x^{3}+3x^{2}+3x+28=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 28 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±28,±14,±7,±4,±2,±1$$

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.

$$x=-4$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+3x^{2}+3x+28 by x+4 to get x^{2}-x+7. Solve the equation where the result equals to 0.

$$x^{2}-x+7=0$$

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, -1 for b, and 7 for c in the quadratic formula.

$$x=\frac{-\left(-1\right)±\sqrt{\left(-1\right)^{2}-4\times 1\times 7}}{2}$$

Do the calculations.

$$x=\frac{1±\sqrt{-27}}{2}$$

Solve the equation x^{2}-x+7=0 when ± is plus and when ± is minus.

$$x=\frac{-3i\sqrt{3}+1}{2} x=\frac{1+3i\sqrt{3}}{2}$$

List all found solutions.

$$x=-4 x=\frac{-3i\sqrt{3}+1}{2} x=\frac{1+3i\sqrt{3}}{2}$$

Solve for x

$$x=-4$$

Solution Steps

Add 28 to both sides.

$$x^{3}+3x^{2}+3x+28=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 28 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±28,±14,±7,±4,±2,±1$$

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.

$$x=-4$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+3x^{2}+3x+28 by x+4 to get x^{2}-x+7. Solve the equation where the result equals to 0.

$$x^{2}-x+7=0$$

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, -1 for b, and 7 for c in the quadratic formula.

$$x=\frac{-\left(-1\right)±\sqrt{\left(-1\right)^{2}-4\times 1\times 7}}{2}$$

Do the calculations.

$$x=\frac{1±\sqrt{-27}}{2}$$

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.

$$x=-4$$