Solve x^3+4x+5=770

$${ x }^{ 3 } +4x+5 = 770$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{259}i-9}{2}\approx -4.5-8.04673847i$$

$$x=9$$

$$x=\frac{-9+\sqrt{259}i}{2}\approx -4.5+8.04673847i$$

Solution Steps

Subtract 770 from both sides.

$$x^{3}+4x+5-770=0$$

Subtract 770 from 5 to get -765.

$$x^{3}+4x-765=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -765 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±765,±255,±153,±85,±51,±45,±17,±15,±9,±5,±3,±1$$

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.

$$x=9$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+4x-765 by x-9 to get x^{2}+9x+85. Solve the equation where the result equals to 0.

$$x^{2}+9x+85=0$$

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, 9 for b, and 85 for c in the quadratic formula.

$$x=\frac{-9±\sqrt{9^{2}-4\times 1\times 85}}{2}$$

Do the calculations.

$$x=\frac{-9±\sqrt{-259}}{2}$$

Solve the equation x^{2}+9x+85=0 when ± is plus and when ± is minus.

$$x=\frac{-\sqrt{259}i-9}{2} x=\frac{-9+\sqrt{259}i}{2}$$

List all found solutions.

$$x=9 x=\frac{-\sqrt{259}i-9}{2} x=\frac{-9+\sqrt{259}i}{2}$$

Solve for x

$$x=9$$

Solution Steps

Subtract 770 from both sides.

$$x^{3}+4x+5-770=0$$

Subtract 770 from 5 to get -765.

$$x^{3}+4x-765=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -765 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±765,±255,±153,±85,±51,±45,±17,±15,±9,±5,±3,±1$$

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.

$$x=9$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+4x-765 by x-9 to get x^{2}+9x+85. Solve the equation where the result equals to 0.

$$x^{2}+9x+85=0$$

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, 9 for b, and 85 for c in the quadratic formula.

$$x=\frac{-9±\sqrt{9^{2}-4\times 1\times 85}}{2}$$

Do the calculations.

$$x=\frac{-9±\sqrt{-259}}{2}$$

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.

$$x=9$$