Solve 1/100=1/t-480+1/t

$$\frac{ 1 }{ 100 } = \frac{ 1 }{ t-480 } + \frac{ 1 }{ t }$$

Solve for t

$$t=80$$

$$t=600$$

Steps Using the Quadratic Formula

Variable t cannot be equal to any of the values 0,480 since division by zero is not defined. Multiply both sides of the equation by 100t\left(t-480\right), the least common multiple of 100,t-480,t.

$$t\left(t-480\right)=100t+100t-48000$$

Use the distributive property to multiply t by t-480.

$$t^{2}-480t=100t+100t-48000$$

Combine 100t and 100t to get 200t.

$$t^{2}-480t=200t-48000$$

Subtract 200t from both sides.

$$t^{2}-480t-200t=-48000$$

Combine -480t and -200t to get -680t.

$$t^{2}-680t=-48000$$

Add 48000 to both sides.

$$t^{2}-680t+48000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -680 for b, and 48000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-680\right)±\sqrt{\left(-680\right)^{2}-4\times 48000}}{2}$$

Square -680.

$$t=\frac{-\left(-680\right)±\sqrt{462400-4\times 48000}}{2}$$

Multiply -4 times 48000.

$$t=\frac{-\left(-680\right)±\sqrt{462400-192000}}{2}$$

Add 462400 to -192000.

$$t=\frac{-\left(-680\right)±\sqrt{270400}}{2}$$

Take the square root of 270400.

$$t=\frac{-\left(-680\right)±520}{2}$$

The opposite of -680 is 680.

$$t=\frac{680±520}{2}$$

Now solve the equation t=\frac{680±520}{2} when ± is plus. Add 680 to 520.

$$t=\frac{1200}{2}$$

Divide 1200 by 2.

$$t=600$$

Now solve the equation t=\frac{680±520}{2} when ± is minus. Subtract 520 from 680.

$$t=\frac{160}{2}$$

Divide 160 by 2.

$$t=80$$

The equation is now solved.

$$t=600 t=80$$

Steps for Completing the Square

Variable t cannot be equal to any of the values 0,480 since division by zero is not defined. Multiply both sides of the equation by 100t\left(t-480\right), the least common multiple of 100,t-480,t.

$$t\left(t-480\right)=100t+100t-48000$$

Use the distributive property to multiply t by t-480.

$$t^{2}-480t=100t+100t-48000$$

Combine 100t and 100t to get 200t.

$$t^{2}-480t=200t-48000$$

Subtract 200t from both sides.

$$t^{2}-480t-200t=-48000$$

Combine -480t and -200t to get -680t.

$$t^{2}-680t=-48000$$

Divide -680, the coefficient of the x term, by 2 to get -340. Then add the square of -340 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-680t+\left(-340\right)^{2}=-48000+\left(-340\right)^{2}$$

Square -340.

$$t^{2}-680t+115600=-48000+115600$$

Add -48000 to 115600.

$$t^{2}-680t+115600=67600$$

Factor t^{2}-680t+115600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-340\right)^{2}=67600$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-340\right)^{2}}=\sqrt{67600}$$

Simplify.

$$t-340=260 t-340=-260$$

Add 340 to both sides of the equation.

$$t=600 t=80$$