Solve 6x^3-7x^2-7x+6=0

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

Solve for x

$$x=\frac{2}{3}\approx 0.666666667$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$x=-1$$

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

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

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 6x^{3}-7x^{2}-7x+6 by x+1 to get 6x^{2}-13x+6. Solve the equation where the result equals to 0.

$$6x^{2}-13x+6=0$$

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 6 for a, -13 for b, and 6 for c in the quadratic formula.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 6\times 6}}{2\times 6}$$

Do the calculations.

$$x=\frac{13±5}{12}$$

Solve the equation 6x^{2}-13x+6=0 when ± is plus and when ± is minus.

$$x=\frac{2}{3} x=\frac{3}{2}$$

List all found solutions.

$$x=-1 x=\frac{2}{3} x=\frac{3}{2}$$