Solve 5t^2-4/13t+4/169-4=0

$$5 t ^ { 2 } - \frac { 4 } { 13 } t + \frac { 4 } { 169 } - 4 = 0$$

Solve for t

$$t=-\frac{56}{65}\approx -0.861538462$$

$$t=\frac{12}{13}\approx 0.923076923$$

Steps Using the Quadratic Formula

Subtract 4 from \frac{4}{169} to get -\frac{672}{169}.

$$5t^{2}-\frac{4}{13}t-\frac{672}{169}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -\frac{4}{13} for b, and -\frac{672}{169} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-\frac{4}{13}\right)±\sqrt{\left(-\frac{4}{13}\right)^{2}-4\times 5\left(-\frac{672}{169}\right)}}{2\times 5}$$

Square -\frac{4}{13} by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\left(-\frac{4}{13}\right)±\sqrt{\frac{16}{169}-4\times 5\left(-\frac{672}{169}\right)}}{2\times 5}$$

Multiply -4 times 5.

$$t=\frac{-\left(-\frac{4}{13}\right)±\sqrt{\frac{16}{169}-20\left(-\frac{672}{169}\right)}}{2\times 5}$$

Multiply -20 times -\frac{672}{169}.

$$t=\frac{-\left(-\frac{4}{13}\right)±\sqrt{\frac{16+13440}{169}}}{2\times 5}$$

Add \frac{16}{169} to \frac{13440}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\left(-\frac{4}{13}\right)±\sqrt{\frac{13456}{169}}}{2\times 5}$$

Take the square root of \frac{13456}{169}.

$$t=\frac{-\left(-\frac{4}{13}\right)±\frac{116}{13}}{2\times 5}$$

The opposite of -\frac{4}{13} is \frac{4}{13}.

$$t=\frac{\frac{4}{13}±\frac{116}{13}}{2\times 5}$$

Multiply 2 times 5.

$$t=\frac{\frac{4}{13}±\frac{116}{13}}{10}$$

Now solve the equation t=\frac{\frac{4}{13}±\frac{116}{13}}{10} when ± is plus. Add \frac{4}{13} to \frac{116}{13} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{\frac{120}{13}}{10}$$

Divide \frac{120}{13} by 10.

$$t=\frac{12}{13}$$

Now solve the equation t=\frac{\frac{4}{13}±\frac{116}{13}}{10} when ± is minus. Subtract \frac{116}{13} from \frac{4}{13} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$t=-\frac{\frac{112}{13}}{10}$$

Divide -\frac{112}{13} by 10.

$$t=-\frac{56}{65}$$

The equation is now solved.

$$t=\frac{12}{13} t=-\frac{56}{65}$$

Steps for Completing the Square

Subtract 4 from \frac{4}{169} to get -\frac{672}{169}.

$$5t^{2}-\frac{4}{13}t-\frac{672}{169}=0$$

Add \frac{672}{169} to both sides. Anything plus zero gives itself.

$$5t^{2}-\frac{4}{13}t=\frac{672}{169}$$

Divide both sides by 5.

$$\frac{5t^{2}-\frac{4}{13}t}{5}=\frac{\frac{672}{169}}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$t^{2}+\left(-\frac{\frac{4}{13}}{5}\right)t=\frac{\frac{672}{169}}{5}$$

Divide -\frac{4}{13} by 5.

$$t^{2}-\frac{4}{65}t=\frac{\frac{672}{169}}{5}$$

Divide \frac{672}{169} by 5.

$$t^{2}-\frac{4}{65}t=\frac{672}{845}$$

Divide -\frac{4}{65}, the coefficient of the x term, by 2 to get -\frac{2}{65}. Then add the square of -\frac{2}{65} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{4}{65}t+\left(-\frac{2}{65}\right)^{2}=\frac{672}{845}+\left(-\frac{2}{65}\right)^{2}$$

Square -\frac{2}{65} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{4}{65}t+\frac{4}{4225}=\frac{672}{845}+\frac{4}{4225}$$

Add \frac{672}{845} to \frac{4}{4225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{4}{65}t+\frac{4}{4225}=\frac{3364}{4225}$$

Factor t^{2}-\frac{4}{65}t+\frac{4}{4225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{2}{65}\right)^{2}=\frac{3364}{4225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{2}{65}\right)^{2}}=\sqrt{\frac{3364}{4225}}$$

Simplify.

$$t-\frac{2}{65}=\frac{58}{65} t-\frac{2}{65}=-\frac{58}{65}$$

Add \frac{2}{65} to both sides of the equation.

$$t=\frac{12}{13} t=-\frac{56}{65}$$