Solve 30x^3+109x^2=196

$$30 { x }^{ 3 } +109 { x }^{ 2 } =196$$

Solve for x

$$x = \frac{7}{6} = 1\frac{1}{6} \approx 1.166666667$$

$$x=-2$$

$$x = -\frac{14}{5} = -2\frac{4}{5} = -2.8$$

Solution Steps

Subtract 196 from both sides.

$$30x^{3}+109x^{2}-196=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -196 and q divides the leading coefficient 30. List all candidates \frac{p}{q}.

$$±\frac{98}{15},±\frac{196}{15},±\frac{98}{5},±\frac{98}{3},±\frac{196}{5},±\frac{196}{3},±98,±196,±\frac{49}{15},±\frac{49}{5},±\frac{49}{3},±49,±\frac{49}{30},±\frac{49}{10},±\frac{49}{6},±\frac{49}{2},±\frac{14}{15},±\frac{28}{15},±\frac{14}{5},±\frac{14}{3},±\frac{28}{5},±\frac{28}{3},±14,±28,±\frac{7}{15},±\frac{7}{5},±\frac{7}{3},±7,±\frac{7}{30},±\frac{7}{10},±\frac{7}{6},±\frac{7}{2},±\frac{2}{15},±\frac{4}{15},±\frac{2}{5},±\frac{2}{3},±\frac{4}{5},±\frac{4}{3},±2,±4,±\frac{1}{15},±\frac{1}{5},±\frac{1}{3},±1,±\frac{1}{30},±\frac{1}{10},±\frac{1}{6},±\frac{1}{2}$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 30x^{3}+109x^{2}-196 by x+2 to get 30x^{2}+49x-98. Solve the equation where the result equals to 0.

$$30x^{2}+49x-98=0$$

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 30 for a, 49 for b, and -98 for c in the quadratic formula.

$$x=\frac{-49±\sqrt{49^{2}-4\times 30\left(-98\right)}}{2\times 30}$$

Do the calculations.

$$x=\frac{-49±119}{60}$$

Solve the equation 30x^{2}+49x-98=0 when ± is plus and when ± is minus.

$$x=-\frac{14}{5} x=\frac{7}{6}$$

List all found solutions.

$$x=-2 x=-\frac{14}{5} x=\frac{7}{6}$$