Solve x^3+x^2+2x+40-x=124

$$x ^ { 3 } + x ^ { 2 } + 2 x + 40 - x = 124$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{59}i-5}{2}\approx -2.5-3.840572874i$$

$$x=4$$

$$x=\frac{-5+\sqrt{59}i}{2}\approx -2.5+3.840572874i$$

Solution Steps

Combine 2x and -x to get x.

$$x^{3}+x^{2}+x+40=124$$

Subtract 124 from both sides.

$$x^{3}+x^{2}+x+40-124=0$$

Subtract 124 from 40 to get -84.

$$x^{3}+x^{2}+x-84=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -84 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±84,±42,±28,±21,±14,±12,±7,±6,±4,±3,±2,±1$$

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.

$$x=4$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+x^{2}+x-84 by x-4 to get x^{2}+5x+21. Solve the equation where the result equals to 0.

$$x^{2}+5x+21=0$$

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, 5 for b, and 21 for c in the quadratic formula.

$$x=\frac{-5±\sqrt{5^{2}-4\times 1\times 21}}{2}$$

Do the calculations.

$$x=\frac{-5±\sqrt{-59}}{2}$$

Solve the equation x^{2}+5x+21=0 when ± is plus and when ± is minus.

$$x=\frac{-\sqrt{59}i-5}{2} x=\frac{-5+\sqrt{59}i}{2}$$

List all found solutions.

$$x=4 x=\frac{-\sqrt{59}i-5}{2} x=\frac{-5+\sqrt{59}i}{2}$$

Solve for x

$$x=4$$

Solution Steps

Combine 2x and -x to get x.

$$x^{3}+x^{2}+x+40=124$$

Subtract 124 from both sides.

$$x^{3}+x^{2}+x+40-124=0$$

Subtract 124 from 40 to get -84.

$$x^{3}+x^{2}+x-84=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -84 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±84,±42,±28,±21,±14,±12,±7,±6,±4,±3,±2,±1$$

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.

$$x=4$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+x^{2}+x-84 by x-4 to get x^{2}+5x+21. Solve the equation where the result equals to 0.

$$x^{2}+5x+21=0$$

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, 5 for b, and 21 for c in the quadratic formula.

$$x=\frac{-5±\sqrt{5^{2}-4\times 1\times 21}}{2}$$

Do the calculations.

$$x=\frac{-5±\sqrt{-59}}{2}$$

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.

$$x=4$$