Solve x^3+10x=57

$$x ^ { 3 } + 10 x = 57$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{67}i-3}{2}\approx -1.5-4.092676386i$$

$$x=3$$

$$x=\frac{-3+\sqrt{67}i}{2}\approx -1.5+4.092676386i$$

Solution Steps

Subtract 57 from both sides.

$$x^{3}+10x-57=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -57 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±57,±19,±3,±1$$

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+10x-57 by x-3 to get x^{2}+3x+19. Solve the equation where the result equals to 0.

$$x^{2}+3x+19=0$$

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, 3 for b, and 19 for c in the quadratic formula.

$$x=\frac{-3±\sqrt{3^{2}-4\times 1\times 19}}{2}$$

Do the calculations.

$$x=\frac{-3±\sqrt{-67}}{2}$$

Solve the equation x^{2}+3x+19=0 when ± is plus and when ± is minus.

$$x=\frac{-\sqrt{67}i-3}{2} x=\frac{-3+\sqrt{67}i}{2}$$

List all found solutions.

$$x=3 x=\frac{-\sqrt{67}i-3}{2} x=\frac{-3+\sqrt{67}i}{2}$$

Solve for x

$$x=3$$

Solution Steps

Subtract 57 from both sides.

$$x^{3}+10x-57=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -57 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±57,±19,±3,±1$$

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+10x-57 by x-3 to get x^{2}+3x+19. Solve the equation where the result equals to 0.

$$x^{2}+3x+19=0$$

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, 3 for b, and 19 for c in the quadratic formula.

$$x=\frac{-3±\sqrt{3^{2}-4\times 1\times 19}}{2}$$

Do the calculations.

$$x=\frac{-3±\sqrt{-67}}{2}$$

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.

$$x=3$$