Solve 3x^3-4x^2-13x-6=0

$$3 { x }^{ 3 } -4 { x }^{ 2 } -13x-6=0$$

Solve for x

$$x=-1$$

$$x=3$$

$$x=-\frac{2}{3}\approx -0.666666667$$

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 -6 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±2,±6,±1,±3,±\frac{2}{3},±\frac{1}{3}$$

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 3x^{3}-4x^{2}-13x-6 by x+1 to get 3x^{2}-7x-6. Solve the equation where the result equals to 0.

$$3x^{2}-7x-6=0$$

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, -7 for b, and -6 for c in the quadratic formula.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 3\left(-6\right)}}{2\times 3}$$

Do the calculations.

$$x=\frac{7±11}{6}$$

Solve the equation 3x^{2}-7x-6=0 when ± is plus and when ± is minus.

$$x=-\frac{2}{3} x=3$$

List all found solutions.

$$x=-1 x=-\frac{2}{3} x=3$$