Solve x^3-5x^2+2x+2=0

$${ x }^{ 3 } -5 { x }^{ 2 } +2x+2 = 0$$

Solve for x

$$x=\sqrt{6}+2\approx 4.449489743$$

$$x=2-\sqrt{6}\approx -0.449489743$$

$$x=1$$

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

$$±2,±1$$

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.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-5x^{2}+2x+2 by x-1 to get x^{2}-4x-2. Solve the equation where the result equals to 0.

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

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

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

Do the calculations.

$$x=\frac{4±2\sqrt{6}}{2}$$

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

$$x=2-\sqrt{6} x=\sqrt{6}+2$$

List all found solutions.

$$x=1 x=2-\sqrt{6} x=\sqrt{6}+2$$