Solve 27(x+1)^3-64=0

$$27 ( x + 1 ) ^ { 3 } - 64 = 0$$

Solve for x (complex solution)

$$x=\frac{1}{3}\approx 0.333333333$$

$$x=\frac{-5+2\sqrt{3}i}{3}\approx -1.666666667+1.154700538i$$

$$x=\frac{-2\sqrt{3}i-5}{3}\approx -1.666666667-1.154700538i$$

Solution Steps

Expand the expression.

$$27x^{3}+81x^{2}+81x-37=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -37 and q divides the leading coefficient 27. List all candidates \frac{p}{q}.

$$±\frac{37}{27},±\frac{37}{9},±\frac{37}{3},±37,±\frac{1}{27},±\frac{1}{9},±\frac{1}{3},±1$$

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.

$$x=\frac{1}{3}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 27x^{3}+81x^{2}+81x-37 by 3\left(x-\frac{1}{3}\right)=3x-1 to get 9x^{2}+30x+37. Solve the equation where the result equals to 0.

$$9x^{2}+30x+37=0$$

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, 30 for b, and 37 for c in the quadratic formula.

$$x=\frac{-30±\sqrt{30^{2}-4\times 9\times 37}}{2\times 9}$$

Do the calculations.

$$x=\frac{-30±\sqrt{-432}}{18}$$

Solve the equation 9x^{2}+30x+37=0 when ± is plus and when ± is minus.

$$x=\frac{-2i\sqrt{3}-5}{3} x=\frac{-5+2i\sqrt{3}}{3}$$

List all found solutions.

$$x=\frac{1}{3} x=\frac{-2i\sqrt{3}-5}{3} x=\frac{-5+2i\sqrt{3}}{3}$$

Solve for x

$$x=\frac{1}{3}\approx 0.333333333$$

Solution Steps

Expand the expression.

$$27x^{3}+81x^{2}+81x-37=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -37 and q divides the leading coefficient 27. List all candidates \frac{p}{q}.

$$±\frac{37}{27},±\frac{37}{9},±\frac{37}{3},±37,±\frac{1}{27},±\frac{1}{9},±\frac{1}{3},±1$$

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.

$$x=\frac{1}{3}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 27x^{3}+81x^{2}+81x-37 by 3\left(x-\frac{1}{3}\right)=3x-1 to get 9x^{2}+30x+37. Solve the equation where the result equals to 0.

$$9x^{2}+30x+37=0$$

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, 30 for b, and 37 for c in the quadratic formula.

$$x=\frac{-30±\sqrt{30^{2}-4\times 9\times 37}}{2\times 9}$$

Do the calculations.

$$x=\frac{-30±\sqrt{-432}}{18}$$

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.

$$x=\frac{1}{3}$$