Solve 1-t^3/1-t=7/5

$$\frac { 1 - t ^ { 3 } } { 1 - t } = \frac { 7 } { 5 }$$

Solve for t

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

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

Solution Steps

Variable t cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 5\left(t-1\right), the least common multiple of 1-t,5.

$$-5\left(1-t^{3}\right)=7\left(t-1\right)$$

Use the distributive property to multiply -5 by 1-t^{3}.

$$-5+5t^{3}=7\left(t-1\right)$$

Use the distributive property to multiply 7 by t-1.

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

Subtract 7t from both sides.

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

Add 7 to both sides.

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

Add -5 and 7 to get 2.

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

Rearrange the equation to put it in standard form. Place the terms in order from highest to lowest power.

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

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.

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

Remove the values that the variable cannot be equal to.

$$t\in \emptyset$$

List all found solutions.

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

Variable t cannot be equal to 1.

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