Solve 5x^3+6x^2+x-66=0

$$5 x ^ { 3 } + 6 x ^ { 2 } + x - 66 = 0$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{101}i-8}{5}\approx -1.6-2.009975124i$$

$$x=2$$

$$x=\frac{-8+\sqrt{101}i}{5}\approx -1.6+2.009975124i$$

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 -66 and q divides the leading coefficient 5. List all candidates \frac{p}{q}.

$$±\frac{66}{5},±66,±\frac{33}{5},±33,±\frac{22}{5},±22,±\frac{11}{5},±11,±\frac{6}{5},±6,±\frac{3}{5},±3,±\frac{2}{5},±2,±\frac{1}{5},±1$$

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 5x^{3}+6x^{2}+x-66 by x-2 to get 5x^{2}+16x+33. Solve the equation where the result equals to 0.

$$5x^{2}+16x+33=0$$

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 5 for a, 16 for b, and 33 for c in the quadratic formula.

$$x=\frac{-16±\sqrt{16^{2}-4\times 5\times 33}}{2\times 5}$$

Do the calculations.

$$x=\frac{-16±\sqrt{-404}}{10}$$

Solve the equation 5x^{2}+16x+33=0 when ± is plus and when ± is minus.

$$x=\frac{-\sqrt{101}i-8}{5} x=\frac{-8+\sqrt{101}i}{5}$$

List all found solutions.

$$x=2 x=\frac{-\sqrt{101}i-8}{5} x=\frac{-8+\sqrt{101}i}{5}$$

Solve for x

$$x=2$$

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 -66 and q divides the leading coefficient 5. List all candidates \frac{p}{q}.

$$±\frac{66}{5},±66,±\frac{33}{5},±33,±\frac{22}{5},±22,±\frac{11}{5},±11,±\frac{6}{5},±6,±\frac{3}{5},±3,±\frac{2}{5},±2,±\frac{1}{5},±1$$

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 5x^{3}+6x^{2}+x-66 by x-2 to get 5x^{2}+16x+33. Solve the equation where the result equals to 0.

$$5x^{2}+16x+33=0$$

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 5 for a, 16 for b, and 33 for c in the quadratic formula.

$$x=\frac{-16±\sqrt{16^{2}-4\times 5\times 33}}{2\times 5}$$

Do the calculations.

$$x=\frac{-16±\sqrt{-404}}{10}$$

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.

$$x=2$$