Solve x^3-8x^2+8x+24=0

$${ x }^{ 3 } -8 { x }^{ 2 } +8x+24 = 0$$

Solve for x

$$x=1-\sqrt{5}\approx -1.236067977$$

$$x=6$$

$$x=\sqrt{5}+1\approx 3.236067977$$

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

$$±24,±12,±8,±6,±4,±3,±2,±1$$

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.

$$x=6$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-8x^{2}+8x+24 by x-6 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.

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

Do the calculations.

$$x=\frac{2±2\sqrt{5}}{2}$$

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

$$x=1-\sqrt{5} x=\sqrt{5}+1$$

List all found solutions.

$$x=6 x=1-\sqrt{5} x=\sqrt{5}+1$$