Solve 5t^3-7t+2=0

$$5 t ^ { 3 } - 7 t + 2 = 0$$

Solve for t

$$t=\frac{\sqrt{65}}{10}-\frac{1}{2}\approx 0.306225775$$

$$t=-\frac{\sqrt{65}}{10}-\frac{1}{2}\approx -1.306225775$$

$$t=1$$

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

$$±\frac{2}{5},±2,±\frac{1}{5},±1$$

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.

$$t=1$$

By Factor theorem, t-k is a factor of the polynomial for each root k. Divide 5t^{3}-7t+2 by t-1 to get 5t^{2}+5t-2. Solve the equation where the result equals to 0.

$$5t^{2}+5t-2=0$$

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 5 for a, 5 for b, and -2 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-5±\sqrt{65}}{10}$$

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

$$t=-\frac{\sqrt{65}}{10}-\frac{1}{2} t=\frac{\sqrt{65}}{10}-\frac{1}{2}$$

List all found solutions.

$$t=1 t=-\frac{\sqrt{65}}{10}-\frac{1}{2} t=\frac{\sqrt{65}}{10}-\frac{1}{2}$$