Solve x^3-9x^2+64=0

$$x ^ { 3 } - 9 x ^ { 2 } + 64 = 0$$

Solve for x

$$x = \frac{\sqrt{33} + 1}{2} \approx 3.372281323$$

$$x=\frac{1-\sqrt{33}}{2}\approx -2.372281323$$

$$x=8$$

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.

$$x=8$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-9x^{2}+64 by x-8 to get x^{2}-x-8. Solve the equation where the result equals to 0.

$$x^{2}-x-8=0$$

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, -1 for b, and -8 for c in the quadratic formula.

$$x=\frac{-\left(-1\right)±\sqrt{\left(-1\right)^{2}-4\times 1\left(-8\right)}}{2}$$

Do the calculations.

$$x=\frac{1±\sqrt{33}}{2}$$

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

$$x=\frac{1-\sqrt{33}}{2} x=\frac{\sqrt{33}+1}{2}$$

List all found solutions.

$$x=8 x=\frac{1-\sqrt{33}}{2} x=\frac{\sqrt{33}+1}{2}$$