Solve 2q^3-3q^2+1=0

$$2 q ^ { 3 } - 3 q ^ { 2 } + 1 = 0$$

Solve for q

$$q=1$$

$$q=-\frac{1}{2}=-0.5$$

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 1 and q divides the leading coefficient 2. List all candidates \frac{p}{q}.

$$±\frac{1}{2},±1$$

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.

$$q=1$$

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

$$2q^{2}-q-1=0$$

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 2 for a, -1 for b, and -1 for c in the quadratic formula.

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

Do the calculations.

$$q=\frac{1±3}{4}$$

Solve the equation 2q^{2}-q-1=0 when ± is plus and when ± is minus.

$$q=-\frac{1}{2} q=1$$

List all found solutions.

$$q=1 q=-\frac{1}{2}$$