Solve 9x^3-12x^2-8x-1=0

$$9 x ^ { 3 } - 12 x ^ { 2 } - 8 x - 1 = 0$$

Solve for x

$$x = \frac{\sqrt{37} + 5}{6} \approx 1.847127088$$

$$x=\frac{5-\sqrt{37}}{6}\approx -0.180460422$$

$$x=-\frac{1}{3}\approx -0.333333333$$

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 -1 and q divides the leading coefficient 9. List all candidates \frac{p}{q}.

$$±\frac{1}{9},±\frac{1}{3},±1$$

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.

$$x=-\frac{1}{3}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 9x^{3}-12x^{2}-8x-1 by 3\left(x+\frac{1}{3}\right)=3x+1 to get 3x^{2}-5x-1. Solve the equation where the result equals to 0.

$$3x^{2}-5x-1=0$$

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 3 for a, -5 for b, and -1 for c in the quadratic formula.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\left(-1\right)}}{2\times 3}$$

Do the calculations.

$$x=\frac{5±\sqrt{37}}{6}$$

Solve the equation 3x^{2}-5x-1=0 when ± is plus and when ± is minus.

$$x=\frac{5-\sqrt{37}}{6} x=\frac{\sqrt{37}+5}{6}$$

List all found solutions.

$$x=-\frac{1}{3} x=\frac{5-\sqrt{37}}{6} x=\frac{\sqrt{37}+5}{6}$$