Solve 2x^3-3x^2-7x-6=0

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

Solve for x (complex solution)

$$x=\frac{-\sqrt{7}i-3}{4}\approx -0.75-0.661437828i$$

$$x=3$$

$$x=\frac{-3+\sqrt{7}i}{4}\approx -0.75+0.661437828i$$

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 2. List all candidates \frac{p}{q}.

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

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 2x^{3}-3x^{2}-7x-6 by x-3 to get 2x^{2}+3x+2. Solve the equation where the result equals to 0.

$$2x^{2}+3x+2=0$$

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 2 for a, 3 for b, and 2 for c in the quadratic formula.

$$x=\frac{-3±\sqrt{3^{2}-4\times 2\times 2}}{2\times 2}$$

Do the calculations.

$$x=\frac{-3±\sqrt{-7}}{4}$$

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

$$x=\frac{-\sqrt{7}i-3}{4} x=\frac{-3+\sqrt{7}i}{4}$$

List all found solutions.

$$x=3 x=\frac{-\sqrt{7}i-3}{4} x=\frac{-3+\sqrt{7}i}{4}$$

Solve for x

$$x=3$$

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 2. List all candidates \frac{p}{q}.

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

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 2x^{3}-3x^{2}-7x-6 by x-3 to get 2x^{2}+3x+2. Solve the equation where the result equals to 0.

$$2x^{2}+3x+2=0$$

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 2 for a, 3 for b, and 2 for c in the quadratic formula.

$$x=\frac{-3±\sqrt{3^{2}-4\times 2\times 2}}{2\times 2}$$

Do the calculations.

$$x=\frac{-3±\sqrt{-7}}{4}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$x\in \emptyset$$

List all found solutions.

$$x=3$$