Solve x^3+7x^2+14x-64=0

$$x ^ { 3 } + 7 x ^ { 2 } + 14 x - 64 = 0$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{47}i-9}{2}\approx -4.5-3.4278273i$$

$$x=2$$

$$x=\frac{-9+\sqrt{47}i}{2}\approx -4.5+3.4278273i$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -64 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±64,±32,±16,±8,±4,±2,±1$$

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+7x^{2}+14x-64 by x-2 to get x^{2}+9x+32. Solve the equation where the result equals to 0.

$$x^{2}+9x+32=0$$

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 32 for c in the quadratic formula.

$$x=\frac{-9±\sqrt{9^{2}-4\times 1\times 32}}{2}$$

Do the calculations.

$$x=\frac{-9±\sqrt{-47}}{2}$$

Solve the equation x^{2}+9x+32=0 when ± is plus and when ± is minus.

$$x=\frac{-\sqrt{47}i-9}{2} x=\frac{-9+\sqrt{47}i}{2}$$

List all found solutions.

$$x=2 x=\frac{-\sqrt{47}i-9}{2} x=\frac{-9+\sqrt{47}i}{2}$$

Solve for x

$$x=2$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -64 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±64,±32,±16,±8,±4,±2,±1$$

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+7x^{2}+14x-64 by x-2 to get x^{2}+9x+32. Solve the equation where the result equals to 0.

$$x^{2}+9x+32=0$$

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 32 for c in the quadratic formula.

$$x=\frac{-9±\sqrt{9^{2}-4\times 1\times 32}}{2}$$

Do the calculations.

$$x=\frac{-9±\sqrt{-47}}{2}$$

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.

$$x=2$$