Solve 24/t-20-24/t+20=1/3

$$\frac { 24 } { t - 20 } - \frac { 24 } { t + 20 } = \frac { 1 } { 3 }$$

Solve for t

$$t=4\sqrt{205}\approx 57.271284253$$

$$t=-4\sqrt{205}\approx -57.271284253$$

Steps by Finding Square Root

Variable t cannot be equal to any of the values -20,20 since division by zero is not defined. Multiply both sides of the equation by 3\left(t-20\right)\left(t+20\right), the least common multiple of t-20,t+20,3.

$$\left(3t+60\right)\times 24-\left(3t-60\right)\times 24=\left(t-20\right)\left(t+20\right)$$

Use the distributive property to multiply 3t+60 by 24.

$$72t+1440-\left(3t-60\right)\times 24=\left(t-20\right)\left(t+20\right)$$

Use the distributive property to multiply 3t-60 by 24.

$$72t+1440-\left(72t-1440\right)=\left(t-20\right)\left(t+20\right)$$

To find the opposite of 72t-1440, find the opposite of each term.

$$72t+1440-72t+1440=\left(t-20\right)\left(t+20\right)$$

Combine 72t and -72t to get 0.

$$1440+1440=\left(t-20\right)\left(t+20\right)$$

Add 1440 and 1440 to get 2880.

$$2880=\left(t-20\right)\left(t+20\right)$$

Consider \left(t-20\right)\left(t+20\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 20.

$$2880=t^{2}-400$$

Swap sides so that all variable terms are on the left hand side.

$$t^{2}-400=2880$$

Add 400 to both sides.

$$t^{2}=2880+400$$

Add 2880 and 400 to get 3280.

$$t^{2}=3280$$

Take the square root of both sides of the equation.

$$t=4\sqrt{205} t=-4\sqrt{205}$$

Steps Using the Quadratic Formula

Variable t cannot be equal to any of the values -20,20 since division by zero is not defined. Multiply both sides of the equation by 3\left(t-20\right)\left(t+20\right), the least common multiple of t-20,t+20,3.

$$\left(3t+60\right)\times 24-\left(3t-60\right)\times 24=\left(t-20\right)\left(t+20\right)$$

Use the distributive property to multiply 3t+60 by 24.

$$72t+1440-\left(3t-60\right)\times 24=\left(t-20\right)\left(t+20\right)$$

Use the distributive property to multiply 3t-60 by 24.

$$72t+1440-\left(72t-1440\right)=\left(t-20\right)\left(t+20\right)$$

To find the opposite of 72t-1440, find the opposite of each term.

$$72t+1440-72t+1440=\left(t-20\right)\left(t+20\right)$$

Combine 72t and -72t to get 0.

$$1440+1440=\left(t-20\right)\left(t+20\right)$$

Add 1440 and 1440 to get 2880.

$$2880=\left(t-20\right)\left(t+20\right)$$

Consider \left(t-20\right)\left(t+20\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 20.

$$2880=t^{2}-400$$

Swap sides so that all variable terms are on the left hand side.

$$t^{2}-400=2880$$

Subtract 2880 from both sides.

$$t^{2}-400-2880=0$$

Subtract 2880 from -400 to get -3280.

$$t^{2}-3280=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -3280 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{0±\sqrt{0^{2}-4\left(-3280\right)}}{2}$$

Square 0.

$$t=\frac{0±\sqrt{-4\left(-3280\right)}}{2}$$

Multiply -4 times -3280.

$$t=\frac{0±\sqrt{13120}}{2}$$

Take the square root of 13120.

$$t=\frac{0±8\sqrt{205}}{2}$$

Now solve the equation t=\frac{0±8\sqrt{205}}{2} when ± is plus.

$$t=4\sqrt{205}$$

Now solve the equation t=\frac{0±8\sqrt{205}}{2} when ± is minus.

$$t=-4\sqrt{205}$$

The equation is now solved.

$$t=4\sqrt{205} t=-4\sqrt{205}$$