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

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

Solve for t

$$t=-400$$

$$t=120$$

Steps Using the Quadratic Formula

Variable t cannot be equal to any of the values -480,0 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 280t.

$$t^{2}+280t=48000$$

Subtract 48000 from both sides.

$$t^{2}+280t-48000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 280 for b, and -48000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-280±\sqrt{280^{2}-4\left(-48000\right)}}{2}$$

Square 280.

$$t=\frac{-280±\sqrt{78400-4\left(-48000\right)}}{2}$$

Multiply -4 times -48000.

$$t=\frac{-280±\sqrt{78400+192000}}{2}$$

Add 78400 to 192000.

$$t=\frac{-280±\sqrt{270400}}{2}$$

Take the square root of 270400.

$$t=\frac{-280±520}{2}$$

Now solve the equation t=\frac{-280±520}{2} when ± is plus. Add -280 to 520.

$$t=\frac{240}{2}$$

Divide 240 by 2.

$$t=120$$

Now solve the equation t=\frac{-280±520}{2} when ± is minus. Subtract 520 from -280.

$$t=-\frac{800}{2}$$

Divide -800 by 2.

$$t=-400$$

The equation is now solved.

$$t=120 t=-400$$

Steps for Completing the Square

Variable t cannot be equal to any of the values -480,0 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 280t.

$$t^{2}+280t=48000$$

Divide 280, the coefficient of the x term, by 2 to get 140. Then add the square of 140 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+280t+140^{2}=48000+140^{2}$$

Square 140.

$$t^{2}+280t+19600=48000+19600$$

Add 48000 to 19600.

$$t^{2}+280t+19600=67600$$

Factor t^{2}+280t+19600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+140\right)^{2}=67600$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+140\right)^{2}}=\sqrt{67600}$$

Simplify.

$$t+140=260 t+140=-260$$

Subtract 140 from both sides of the equation.

$$t=120 t=-400$$