Solve 1/t+1/t+3=1/4

$$\frac{ 1 }{ t } + \frac{ 1 }{ t+3 } = \frac{ 1 }{ 4 }$$

Solve for t

$$t = \frac{\sqrt{73} + 5}{2} \approx 6.772001873$$

$$t=\frac{5-\sqrt{73}}{2}\approx -1.772001873$$

Steps Using the Quadratic Formula

Variable t cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by 4t\left(t+3\right), the least common multiple of t,t+3,4.

$$4t+12+4t=t\left(t+3\right)$$

Combine 4t and 4t to get 8t.

$$8t+12=t\left(t+3\right)$$

Use the distributive property to multiply t by t+3.

$$8t+12=t^{2}+3t$$

Subtract t^{2} from both sides.

$$8t+12-t^{2}=3t$$

Subtract 3t from both sides.

$$8t+12-t^{2}-3t=0$$

Combine 8t and -3t to get 5t.

$$5t+12-t^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-t^{2}+5t+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 5 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Square 5.

$$t=\frac{-5±\sqrt{25-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-5±\sqrt{25+4\times 12}}{2\left(-1\right)}$$

Multiply 4 times 12.

$$t=\frac{-5±\sqrt{25+48}}{2\left(-1\right)}$$

Add 25 to 48.

$$t=\frac{-5±\sqrt{73}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{-5±\sqrt{73}}{-2}$$

Now solve the equation t=\frac{-5±\sqrt{73}}{-2} when ± is plus. Add -5 to \sqrt{73}.

$$t=\frac{\sqrt{73}-5}{-2}$$

Divide -5+\sqrt{73} by -2.

$$t=\frac{5-\sqrt{73}}{2}$$

Now solve the equation t=\frac{-5±\sqrt{73}}{-2} when ± is minus. Subtract \sqrt{73} from -5.

$$t=\frac{-\sqrt{73}-5}{-2}$$

Divide -5-\sqrt{73} by -2.

$$t=\frac{\sqrt{73}+5}{2}$$

The equation is now solved.

$$t=\frac{5-\sqrt{73}}{2} t=\frac{\sqrt{73}+5}{2}$$

Steps for Completing the Square

Variable t cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by 4t\left(t+3\right), the least common multiple of t,t+3,4.

$$4t+12+4t=t\left(t+3\right)$$

Combine 4t and 4t to get 8t.

$$8t+12=t\left(t+3\right)$$

Use the distributive property to multiply t by t+3.

$$8t+12=t^{2}+3t$$

Subtract t^{2} from both sides.

$$8t+12-t^{2}=3t$$

Subtract 3t from both sides.

$$8t+12-t^{2}-3t=0$$

Combine 8t and -3t to get 5t.

$$5t+12-t^{2}=0$$

Subtract 12 from both sides. Anything subtracted from zero gives its negation.

$$5t-t^{2}=-12$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-t^{2}+5t=-12$$

Divide both sides by -1.

$$\frac{-t^{2}+5t}{-1}=-\frac{12}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$t^{2}+\frac{5}{-1}t=-\frac{12}{-1}$$

Divide 5 by -1.

$$t^{2}-5t=-\frac{12}{-1}$$

Divide -12 by -1.

$$t^{2}-5t=12$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-5t+\left(-\frac{5}{2}\right)^{2}=12+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-5t+\frac{25}{4}=12+\frac{25}{4}$$

Add 12 to \frac{25}{4}.

$$t^{2}-5t+\frac{25}{4}=\frac{73}{4}$$

Factor t^{2}-5t+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{5}{2}\right)^{2}=\frac{73}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{5}{2}\right)^{2}}=\sqrt{\frac{73}{4}}$$

Simplify.

$$t-\frac{5}{2}=\frac{\sqrt{73}}{2} t-\frac{5}{2}=-\frac{\sqrt{73}}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$t=\frac{\sqrt{73}+5}{2} t=\frac{5-\sqrt{73}}{2}$$