Solve x^3-6x^2+12x-16=0

$${ x }^{ 3 } -6 { x }^{ 2 } +12x-16 = 0$$

Solve for x (complex solution)

$$x=-\sqrt{3}i+1\approx 1-1.732050808i$$

$$x=4$$

$$x=1+\sqrt{3}i\approx 1+1.732050808i$$

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 -16 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±16,±8,±4,±2,±1$$

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.

$$x=4$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-6x^{2}+12x-16 by x-4 to get x^{2}-2x+4. Solve the equation where the result equals to 0.

$$x^{2}-2x+4=0$$

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, -2 for b, and 4 for c in the quadratic formula.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 1\times 4}}{2}$$

Do the calculations.

$$x=\frac{2±\sqrt{-12}}{2}$$

Solve the equation x^{2}-2x+4=0 when ± is plus and when ± is minus.

$$x=-\sqrt{3}i+1 x=1+\sqrt{3}i$$

List all found solutions.

$$x=4 x=-\sqrt{3}i+1 x=1+\sqrt{3}i$$

Solve for x

$$x=4$$

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 -16 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±16,±8,±4,±2,±1$$

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.

$$x=4$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-6x^{2}+12x-16 by x-4 to get x^{2}-2x+4. Solve the equation where the result equals to 0.

$$x^{2}-2x+4=0$$

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, -2 for b, and 4 for c in the quadratic formula.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 1\times 4}}{2}$$

Do the calculations.

$$x=\frac{2±\sqrt{-12}}{2}$$

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.

$$x=4$$