Solve 3x^3-10x^2+9x-30=0

$$3 x ^ { 3 } - 10 x ^ { 2 } + 9 x - 30 = 0$$

Solve for x (complex solution)

$$x=\sqrt{3}i\approx 1.732050808i$$

$$x = \frac{10}{3} = 3\frac{1}{3} \approx 3.333333333$$

$$x=-\sqrt{3}i\approx -0-1.732050808i$$

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 -30 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±10,±30,±5,±15,±\frac{10}{3},±2,±6,±\frac{5}{3},±1,±3,±\frac{2}{3},±\frac{1}{3}$$

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.

$$x=\frac{10}{3}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 3x^{3}-10x^{2}+9x-30 by 3\left(x-\frac{10}{3}\right)=3x-10 to get x^{2}+3. Solve the equation where the result equals to 0.

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

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, 0 for b, and 3 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 1\times 3}}{2}$$

Do the calculations.

$$x=\frac{0±\sqrt{-12}}{2}$$

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

$$x=-\sqrt{3}i x=\sqrt{3}i$$

List all found solutions.

$$x=\frac{10}{3} x=-\sqrt{3}i x=\sqrt{3}i$$

Solve for x

$$x = \frac{10}{3} = 3\frac{1}{3} \approx 3.333333333$$

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 -30 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±10,±30,±5,±15,±\frac{10}{3},±2,±6,±\frac{5}{3},±1,±3,±\frac{2}{3},±\frac{1}{3}$$

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.

$$x=\frac{10}{3}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 3x^{3}-10x^{2}+9x-30 by 3\left(x-\frac{10}{3}\right)=3x-10 to get x^{2}+3. Solve the equation where the result equals to 0.

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

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, 0 for b, and 3 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 1\times 3}}{2}$$

Do the calculations.

$$x=\frac{0±\sqrt{-12}}{2}$$

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.

$$x=\frac{10}{3}$$