Solve 3a^3-11a^2+13a-5=0

$$3 a ^ { 3 } - 11 a ^ { 2 } + 13 a - 5 = 0$$

Solve for a

$$a=1$$

$$a = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

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 -5 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±\frac{5}{3},±5,±\frac{1}{3},±1$$

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.

$$a=1$$

By Factor theorem, a-k is a factor of the polynomial for each root k. Divide 3a^{3}-11a^{2}+13a-5 by a-1 to get 3a^{2}-8a+5. Solve the equation where the result equals to 0.

$$3a^{2}-8a+5=0$$

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 3 for a, -8 for b, and 5 for c in the quadratic formula.

$$a=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 3\times 5}}{2\times 3}$$

Do the calculations.

$$a=\frac{8±2}{6}$$

Solve the equation 3a^{2}-8a+5=0 when ± is plus and when ± is minus.

$$a=1 a=\frac{5}{3}$$

List all found solutions.

$$a=1 a=\frac{5}{3}$$