Solve 20^2+left(1.5tright)^2-left(12+1.5tright)^2/60t=frac{34^2+{left(1.5tright)}^2-{left(30+1.5tright)}^2}{-102t}

$$\frac{ { 20 }^{ 2 } + { \left(1.5t \right) }^{ 2 } - { \left(12+1.5t \right) }^{ 2 } }{ 60t } = \frac{ { 34 }^{ 2 } + { \left(1.5t \right) }^{ 2 } - { \left(30+1.5t \right) }^{ 2 } }{ -102t }$$

Solve for t

$$t = \frac{32}{7} = 4\frac{4}{7} \approx 4.571428571$$

Steps for Solving Linear Equation

Variable t cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1020t, the least common multiple of 60t,-102t.

$$17\left(20^{2}+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Calculate 20 to the power of 2 and get 400.

$$17\left(400+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Expand \left(1.5t\right)^{2}.

$$17\left(400+1.5^{2}t^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Calculate 1.5 to the power of 2 and get 2.25.

$$17\left(400+2.25t^{2}-\left(12+1.5t\right)^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(12+1.5t\right)^{2}.

$$17\left(400+2.25t^{2}-\left(144+36t+2.25t^{2}\right)\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

To find the opposite of 144+36t+2.25t^{2}, find the opposite of each term.

$$17\left(400+2.25t^{2}-144-36t-2.25t^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Subtract 144 from 400 to get 256.

$$17\left(256+2.25t^{2}-36t-2.25t^{2}\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Combine 2.25t^{2} and -2.25t^{2} to get 0.

$$17\left(256-36t\right)=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Use the distributive property to multiply 17 by 256-36t.

$$4352-612t=-10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Calculate 34 to the power of 2 and get 1156.

$$4352-612t=-10\left(1156+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Expand \left(1.5t\right)^{2}.

$$4352-612t=-10\left(1156+1.5^{2}t^{2}-\left(30+1.5t\right)^{2}\right)$$

Calculate 1.5 to the power of 2 and get 2.25.

$$4352-612t=-10\left(1156+2.25t^{2}-\left(30+1.5t\right)^{2}\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(30+1.5t\right)^{2}.

$$4352-612t=-10\left(1156+2.25t^{2}-\left(900+90t+2.25t^{2}\right)\right)$$

To find the opposite of 900+90t+2.25t^{2}, find the opposite of each term.

$$4352-612t=-10\left(1156+2.25t^{2}-900-90t-2.25t^{2}\right)$$

Subtract 900 from 1156 to get 256.

$$4352-612t=-10\left(256+2.25t^{2}-90t-2.25t^{2}\right)$$

Combine 2.25t^{2} and -2.25t^{2} to get 0.

$$4352-612t=-10\left(256-90t\right)$$

Use the distributive property to multiply -10 by 256-90t.

$$4352-612t=-2560+900t$$

Subtract 900t from both sides.

$$4352-612t-900t=-2560$$

Combine -612t and -900t to get -1512t.

$$4352-1512t=-2560$$

Subtract 4352 from both sides.

$$-1512t=-2560-4352$$

Subtract 4352 from -2560 to get -6912.

$$-1512t=-6912$$

Divide both sides by -1512.

$$t=\frac{-6912}{-1512}$$

Reduce the fraction \frac{-6912}{-1512} to lowest terms by extracting and canceling out -216.

$$t=\frac{32}{7}$$