Solve x^3+x^2=392

$$x ^ { 3 } + x ^ { 2 } = 392$$

Solve for x (complex solution)

$$x=-2\sqrt{10}i-4\approx -4-6.32455532i$$

$$x=7$$

$$x=-4+2\sqrt{10}i\approx -4+6.32455532i$$

Solution Steps

Subtract 392 from both sides.

$$x^{3}+x^{2}-392=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -392 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±392,±196,±98,±56,±49,±28,±14,±8,±7,±4,±2,±1$$

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.

$$x=7$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+x^{2}-392 by x-7 to get x^{2}+8x+56. Solve the equation where the result equals to 0.

$$x^{2}+8x+56=0$$

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, 8 for b, and 56 for c in the quadratic formula.

$$x=\frac{-8±\sqrt{8^{2}-4\times 1\times 56}}{2}$$

Do the calculations.

$$x=\frac{-8±\sqrt{-160}}{2}$$

Solve the equation x^{2}+8x+56=0 when ± is plus and when ± is minus.

$$x=-2i\sqrt{10}-4 x=-4+2i\sqrt{10}$$

List all found solutions.

$$x=7 x=-2i\sqrt{10}-4 x=-4+2i\sqrt{10}$$

Solve for x

$$x=7$$

Solution Steps

Subtract 392 from both sides.

$$x^{3}+x^{2}-392=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -392 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±392,±196,±98,±56,±49,±28,±14,±8,±7,±4,±2,±1$$

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.

$$x=7$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+x^{2}-392 by x-7 to get x^{2}+8x+56. Solve the equation where the result equals to 0.

$$x^{2}+8x+56=0$$

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, 8 for b, and 56 for c in the quadratic formula.

$$x=\frac{-8±\sqrt{8^{2}-4\times 1\times 56}}{2}$$

Do the calculations.

$$x=\frac{-8±\sqrt{-160}}{2}$$

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.

$$x=7$$