Solve 2x^3-27x^2+119x-171=0

$$2 x ^ { 3 } - 27 x ^ { 2 } + 119 x - 171 = 0$$

Solve for x

$$x = \frac{9 - \sqrt{5}}{2} \approx 3.381966011$$

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

$$x = \frac{\sqrt{5} + 9}{2} \approx 5.618033989$$

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 -171 and q divides the leading coefficient 2. List all candidates \frac{p}{q}.

$$±\frac{171}{2},±171,±\frac{57}{2},±57,±\frac{19}{2},±19,±\frac{9}{2},±9,±\frac{3}{2},±3,±\frac{1}{2},±1$$

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.

$$x=\frac{9}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 2x^{3}-27x^{2}+119x-171 by 2\left(x-\frac{9}{2}\right)=2x-9 to get x^{2}-9x+19. Solve the equation where the result equals to 0.

$$x^{2}-9x+19=0$$

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 19 for c in the quadratic formula.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 1\times 19}}{2}$$

Do the calculations.

$$x=\frac{9±\sqrt{5}}{2}$$

Solve the equation x^{2}-9x+19=0 when ± is plus and when ± is minus.

$$x=\frac{9-\sqrt{5}}{2} x=\frac{\sqrt{5}+9}{2}$$

List all found solutions.

$$x=\frac{9}{2} x=\frac{9-\sqrt{5}}{2} x=\frac{\sqrt{5}+9}{2}$$