Solve f(x)=x^3-7x^2-14x=6

$$f ( x ) = x ^ { 3 } - 7 x ^ { 2 } - 14 x = 6$$

Solve for x

$$x=\sqrt{22}+4\approx 8.69041576$$

$$x=4-\sqrt{22}\approx -0.69041576$$

$$x=-1$$

Solution Steps

Subtract 6 from both sides.

$$x^{3}-7x^{2}-14x-6=0$$

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 1. List all candidates \frac{p}{q}.

$$±6,±3,±2,±1$$

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.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-7x^{2}-14x-6 by x+1 to get x^{2}-8x-6. Solve the equation where the result equals to 0.

$$x^{2}-8x-6=0$$

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, -8 for b, and -6 for c in the quadratic formula.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 1\left(-6\right)}}{2}$$

Do the calculations.

$$x=\frac{8±2\sqrt{22}}{2}$$

Solve the equation x^{2}-8x-6=0 when ± is plus and when ± is minus.

$$x=4-\sqrt{22} x=\sqrt{22}+4$$

List all found solutions.

$$x=-1 x=4-\sqrt{22} x=\sqrt{22}+4$$