Solve 4x^3+2x^2-15x-6=0

$$4 x ^ { 3 } + 2 x ^ { 2 } - 15 x - 6 = 0$$

Solve for x

$$x = \frac{\sqrt{21} + 3}{4} \approx 1.895643924$$

$$x=\frac{3-\sqrt{21}}{4}\approx -0.395643924$$

$$x=-2$$

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 -6 and q divides the leading coefficient 4. List all candidates \frac{p}{q}.

$$±\frac{3}{2},±3,±6,±\frac{3}{4},±\frac{1}{2},±1,±2,±\frac{1}{4}$$

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 4x^{3}+2x^{2}-15x-6 by x+2 to get 4x^{2}-6x-3. Solve the equation where the result equals to 0.

$$4x^{2}-6x-3=0$$

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 4 for a, -6 for b, and -3 for c in the quadratic formula.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 4\left(-3\right)}}{2\times 4}$$

Do the calculations.

$$x=\frac{6±2\sqrt{21}}{8}$$

Solve the equation 4x^{2}-6x-3=0 when ± is plus and when ± is minus.

$$x=\frac{3-\sqrt{21}}{4} x=\frac{\sqrt{21}+3}{4}$$

List all found solutions.

$$x=-2 x=\frac{3-\sqrt{21}}{4} x=\frac{\sqrt{21}+3}{4}$$