Solve a^3+2a^2-15a-30=0

$$a ^ { 3 } + 2 a ^ { 2 } - 15 a - 30 = 0$$

Solve for a

$$a=\sqrt{15}\approx 3.872983346$$

$$a=-\sqrt{15}\approx -3.872983346$$

$$a=-2$$

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 -30 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±30,±15,±10,±6,±5,±3,±2,±1$$

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.

$$a=-2$$

By Factor theorem, a-k is a factor of the polynomial for each root k. Divide a^{3}+2a^{2}-15a-30 by a+2 to get a^{2}-15. Solve the equation where the result equals to 0.

$$a^{2}-15=0$$

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 -15 for c in the quadratic formula.

$$a=\frac{0±\sqrt{0^{2}-4\times 1\left(-15\right)}}{2}$$

Do the calculations.

$$a=\frac{0±2\sqrt{15}}{2}$$

Solve the equation a^{2}-15=0 when ± is plus and when ± is minus.

$$a=-\sqrt{15} a=\sqrt{15}$$

List all found solutions.

$$a=-2 a=-\sqrt{15} a=\sqrt{15}$$