Solve 343left(x+3right)^3+27=0

$$343 { \left(x+3 \right) }^{ 3 } +27 = 0$$

Solve for x (complex solution)

$$x = -\frac{24}{7} = -3\frac{3}{7} \approx -3.428571429$$

$$x=\frac{-39+3\sqrt{3}i}{14}\approx -2.785714286+0.371153744i$$

$$x=\frac{-3\sqrt{3}i-39}{14}\approx -2.785714286-0.371153744i$$

Solution Steps

Expand the expression.

$$343x^{3}+3087x^{2}+9261x+9288=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 9288 and q divides the leading coefficient 343. List all candidates \frac{p}{q}.

$$±\frac{9288}{343},±\frac{9288}{49},±\frac{9288}{7},±9288,±\frac{4644}{343},±\frac{4644}{49},±\frac{4644}{7},±4644,±\frac{3096}{343},±\frac{3096}{49},±\frac{3096}{7},±3096,±\frac{2322}{343},±\frac{2322}{49},±\frac{2322}{7},±2322,±\frac{1548}{343},±\frac{1548}{49},±\frac{1548}{7},±1548,±\frac{1161}{343},±\frac{1161}{49},±\frac{1161}{7},±1161,±\frac{1032}{343},±\frac{1032}{49},±\frac{1032}{7},±1032,±\frac{774}{343},±\frac{774}{49},±\frac{774}{7},±774,±\frac{516}{343},±\frac{516}{49},±\frac{516}{7},±516,±\frac{387}{343},±\frac{387}{49},±\frac{387}{7},±387,±\frac{344}{343},±\frac{344}{49},±\frac{344}{7},±344,±\frac{258}{343},±\frac{258}{49},±\frac{258}{7},±258,±\frac{216}{343},±\frac{216}{49},±\frac{216}{7},±216,±\frac{172}{343},±\frac{172}{49},±\frac{172}{7},±172,±\frac{129}{343},±\frac{129}{49},±\frac{129}{7},±129,±\frac{108}{343},±\frac{108}{49},±\frac{108}{7},±108,±\frac{86}{343},±\frac{86}{49},±\frac{86}{7},±86,±\frac{72}{343},±\frac{72}{49},±\frac{72}{7},±72,±\frac{54}{343},±\frac{54}{49},±\frac{54}{7},±54,±\frac{43}{343},±\frac{43}{49},±\frac{43}{7},±43,±\frac{36}{343},±\frac{36}{49},±\frac{36}{7},±36,±\frac{27}{343},±\frac{27}{49},±\frac{27}{7},±27,±\frac{24}{343},±\frac{24}{49},±\frac{24}{7},±24,±\frac{18}{343},±\frac{18}{49},±\frac{18}{7},±18,±\frac{12}{343},±\frac{12}{49},±\frac{12}{7},±12,±\frac{9}{343},±\frac{9}{49},±\frac{9}{7},±9,±\frac{8}{343},±\frac{8}{49},±\frac{8}{7},±8,±\frac{6}{343},±\frac{6}{49},±\frac{6}{7},±6,±\frac{4}{343},±\frac{4}{49},±\frac{4}{7},±4,±\frac{3}{343},±\frac{3}{49},±\frac{3}{7},±3,±\frac{2}{343},±\frac{2}{49},±\frac{2}{7},±2,±\frac{1}{343},±\frac{1}{49},±\frac{1}{7},±1$$

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.

$$x=-\frac{24}{7}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 343x^{3}+3087x^{2}+9261x+9288 by 7\left(x+\frac{24}{7}\right)=7x+24 to get 49x^{2}+273x+387. Solve the equation where the result equals to 0.

$$49x^{2}+273x+387=0$$

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 49 for a, 273 for b, and 387 for c in the quadratic formula.

$$x=\frac{-273±\sqrt{273^{2}-4\times 49\times 387}}{2\times 49}$$

Do the calculations.

$$x=\frac{-273±\sqrt{-1323}}{98}$$

Solve the equation 49x^{2}+273x+387=0 when ± is plus and when ± is minus.

$$x=\frac{-3i\sqrt{3}-39}{14} x=\frac{-39+3i\sqrt{3}}{14}$$

List all found solutions.

$$x=-\frac{24}{7} x=\frac{-3i\sqrt{3}-39}{14} x=\frac{-39+3i\sqrt{3}}{14}$$

Solve for x

$$x = -\frac{24}{7} = -3\frac{3}{7} \approx -3.428571429$$

Solution Steps

Expand the expression.

$$343x^{3}+3087x^{2}+9261x+9288=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 9288 and q divides the leading coefficient 343. List all candidates \frac{p}{q}.

$$±\frac{9288}{343},±\frac{9288}{49},±\frac{9288}{7},±9288,±\frac{4644}{343},±\frac{4644}{49},±\frac{4644}{7},±4644,±\frac{3096}{343},±\frac{3096}{49},±\frac{3096}{7},±3096,±\frac{2322}{343},±\frac{2322}{49},±\frac{2322}{7},±2322,±\frac{1548}{343},±\frac{1548}{49},±\frac{1548}{7},±1548,±\frac{1161}{343},±\frac{1161}{49},±\frac{1161}{7},±1161,±\frac{1032}{343},±\frac{1032}{49},±\frac{1032}{7},±1032,±\frac{774}{343},±\frac{774}{49},±\frac{774}{7},±774,±\frac{516}{343},±\frac{516}{49},±\frac{516}{7},±516,±\frac{387}{343},±\frac{387}{49},±\frac{387}{7},±387,±\frac{344}{343},±\frac{344}{49},±\frac{344}{7},±344,±\frac{258}{343},±\frac{258}{49},±\frac{258}{7},±258,±\frac{216}{343},±\frac{216}{49},±\frac{216}{7},±216,±\frac{172}{343},±\frac{172}{49},±\frac{172}{7},±172,±\frac{129}{343},±\frac{129}{49},±\frac{129}{7},±129,±\frac{108}{343},±\frac{108}{49},±\frac{108}{7},±108,±\frac{86}{343},±\frac{86}{49},±\frac{86}{7},±86,±\frac{72}{343},±\frac{72}{49},±\frac{72}{7},±72,±\frac{54}{343},±\frac{54}{49},±\frac{54}{7},±54,±\frac{43}{343},±\frac{43}{49},±\frac{43}{7},±43,±\frac{36}{343},±\frac{36}{49},±\frac{36}{7},±36,±\frac{27}{343},±\frac{27}{49},±\frac{27}{7},±27,±\frac{24}{343},±\frac{24}{49},±\frac{24}{7},±24,±\frac{18}{343},±\frac{18}{49},±\frac{18}{7},±18,±\frac{12}{343},±\frac{12}{49},±\frac{12}{7},±12,±\frac{9}{343},±\frac{9}{49},±\frac{9}{7},±9,±\frac{8}{343},±\frac{8}{49},±\frac{8}{7},±8,±\frac{6}{343},±\frac{6}{49},±\frac{6}{7},±6,±\frac{4}{343},±\frac{4}{49},±\frac{4}{7},±4,±\frac{3}{343},±\frac{3}{49},±\frac{3}{7},±3,±\frac{2}{343},±\frac{2}{49},±\frac{2}{7},±2,±\frac{1}{343},±\frac{1}{49},±\frac{1}{7},±1$$

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.

$$x=-\frac{24}{7}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 343x^{3}+3087x^{2}+9261x+9288 by 7\left(x+\frac{24}{7}\right)=7x+24 to get 49x^{2}+273x+387. Solve the equation where the result equals to 0.

$$49x^{2}+273x+387=0$$

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 49 for a, 273 for b, and 387 for c in the quadratic formula.

$$x=\frac{-273±\sqrt{273^{2}-4\times 49\times 387}}{2\times 49}$$

Do the calculations.

$$x=\frac{-273±\sqrt{-1323}}{98}$$

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.

$$x=-\frac{24}{7}$$