Solve 3^3(x+2)^3+4^3=0

$$3 ^ { 3 } ( x + 2 ) ^ { 3 } + 4 ^ { 3 } = 0$$

Solve for x (complex solution)

$$x = -\frac{10}{3} = -3\frac{1}{3} \approx -3.333333333$$

$$x=\frac{-4+2\sqrt{3}i}{3}\approx -1.333333333+1.154700538i$$

$$x=\frac{-2\sqrt{3}i-4}{3}\approx -1.333333333-1.154700538i$$

Solution Steps

Expand the expression.

$$27x^{3}+162x^{2}+324x+280=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 280 and q divides the leading coefficient 27. List all candidates \frac{p}{q}.

$$±\frac{280}{27},±\frac{280}{9},±\frac{280}{3},±280,±\frac{140}{27},±\frac{140}{9},±\frac{140}{3},±140,±\frac{70}{27},±\frac{70}{9},±\frac{70}{3},±70,±\frac{56}{27},±\frac{56}{9},±\frac{56}{3},±56,±\frac{40}{27},±\frac{40}{9},±\frac{40}{3},±40,±\frac{35}{27},±\frac{35}{9},±\frac{35}{3},±35,±\frac{28}{27},±\frac{28}{9},±\frac{28}{3},±28,±\frac{20}{27},±\frac{20}{9},±\frac{20}{3},±20,±\frac{14}{27},±\frac{14}{9},±\frac{14}{3},±14,±\frac{10}{27},±\frac{10}{9},±\frac{10}{3},±10,±\frac{8}{27},±\frac{8}{9},±\frac{8}{3},±8,±\frac{7}{27},±\frac{7}{9},±\frac{7}{3},±7,±\frac{5}{27},±\frac{5}{9},±\frac{5}{3},±5,±\frac{4}{27},±\frac{4}{9},±\frac{4}{3},±4,±\frac{2}{27},±\frac{2}{9},±\frac{2}{3},±2,±\frac{1}{27},±\frac{1}{9},±\frac{1}{3},±1$$

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.

$$x=-\frac{10}{3}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 27x^{3}+162x^{2}+324x+280 by 3\left(x+\frac{10}{3}\right)=3x+10 to get 9x^{2}+24x+28. Solve the equation where the result equals to 0.

$$9x^{2}+24x+28=0$$

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 9 for a, 24 for b, and 28 for c in the quadratic formula.

$$x=\frac{-24±\sqrt{24^{2}-4\times 9\times 28}}{2\times 9}$$

Do the calculations.

$$x=\frac{-24±\sqrt{-432}}{18}$$

Solve the equation 9x^{2}+24x+28=0 when ± is plus and when ± is minus.

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

List all found solutions.

$$x=-\frac{10}{3} x=\frac{-2i\sqrt{3}-4}{3} x=\frac{-4+2i\sqrt{3}}{3}$$

Solve for x

$$x = -\frac{10}{3} = -3\frac{1}{3} \approx -3.333333333$$

Solution Steps

Expand the expression.

$$27x^{3}+162x^{2}+324x+280=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 280 and q divides the leading coefficient 27. List all candidates \frac{p}{q}.

$$±\frac{280}{27},±\frac{280}{9},±\frac{280}{3},±280,±\frac{140}{27},±\frac{140}{9},±\frac{140}{3},±140,±\frac{70}{27},±\frac{70}{9},±\frac{70}{3},±70,±\frac{56}{27},±\frac{56}{9},±\frac{56}{3},±56,±\frac{40}{27},±\frac{40}{9},±\frac{40}{3},±40,±\frac{35}{27},±\frac{35}{9},±\frac{35}{3},±35,±\frac{28}{27},±\frac{28}{9},±\frac{28}{3},±28,±\frac{20}{27},±\frac{20}{9},±\frac{20}{3},±20,±\frac{14}{27},±\frac{14}{9},±\frac{14}{3},±14,±\frac{10}{27},±\frac{10}{9},±\frac{10}{3},±10,±\frac{8}{27},±\frac{8}{9},±\frac{8}{3},±8,±\frac{7}{27},±\frac{7}{9},±\frac{7}{3},±7,±\frac{5}{27},±\frac{5}{9},±\frac{5}{3},±5,±\frac{4}{27},±\frac{4}{9},±\frac{4}{3},±4,±\frac{2}{27},±\frac{2}{9},±\frac{2}{3},±2,±\frac{1}{27},±\frac{1}{9},±\frac{1}{3},±1$$

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.

$$x=-\frac{10}{3}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 27x^{3}+162x^{2}+324x+280 by 3\left(x+\frac{10}{3}\right)=3x+10 to get 9x^{2}+24x+28. Solve the equation where the result equals to 0.

$$9x^{2}+24x+28=0$$

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 9 for a, 24 for b, and 28 for c in the quadratic formula.

$$x=\frac{-24±\sqrt{24^{2}-4\times 9\times 28}}{2\times 9}$$

Do the calculations.

$$x=\frac{-24±\sqrt{-432}}{18}$$

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.

$$x=-\frac{10}{3}$$