Solve 8x^3-20x^2=-x-14

$$8 { x }^{ 3 } -20 { x }^{ 2 } = -x-14$$

Solve for x

$$x = \frac{\sqrt{15} + 1}{4} \approx 1.218245837$$

$$x=\frac{1-\sqrt{15}}{4}\approx -0.718245837$$

$$x=2$$

Solution Steps

Add x to both sides.

$$8x^{3}-20x^{2}+x=-14$$

Add 14 to both sides.

$$8x^{3}-20x^{2}+x+14=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 14 and q divides the leading coefficient 8. List all candidates \frac{p}{q}.

$$±\frac{7}{4},±\frac{7}{2},±7,±14,±\frac{7}{8},±\frac{1}{4},±\frac{1}{2},±1,±2,±\frac{1}{8}$$

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 8x^{3}-20x^{2}+x+14 by x-2 to get 8x^{2}-4x-7. Solve the equation where the result equals to 0.

$$8x^{2}-4x-7=0$$

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 8 for a, -4 for b, and -7 for c in the quadratic formula.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 8\left(-7\right)}}{2\times 8}$$

Do the calculations.

$$x=\frac{4±4\sqrt{15}}{16}$$

Solve the equation 8x^{2}-4x-7=0 when ± is plus and when ± is minus.

$$x=\frac{1-\sqrt{15}}{4} x=\frac{\sqrt{15}+1}{4}$$

List all found solutions.

$$x=2 x=\frac{1-\sqrt{15}}{4} x=\frac{\sqrt{15}+1}{4}$$