Solve f(x)=2/x^2+400-1/(60-x)^2+1600

$$f ( x ) = \frac { 2 } { x ^ { 2 } + 400 } - \frac { 1 } { ( 60 - x ) ^ { 2 } + 1600 }$$

Evaluate

$$\frac{x^{2}-240x+10000}{\left(x^{2}+400\right)\left(x^{2}-120x+5200\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}+400 and \left(60-x\right)^{2}+1600 is \left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right). Multiply \frac{2}{x^{2}+400} times \frac{\left(-x+60\right)^{2}+1600}{\left(-x+60\right)^{2}+1600}. Multiply \frac{1}{\left(60-x\right)^{2}+1600} times \frac{x^{2}+400}{x^{2}+400}.

$$\frac{2\left(\left(-x+60\right)^{2}+1600\right)}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)}-\frac{x^{2}+400}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)}$$

Since \frac{2\left(\left(-x+60\right)^{2}+1600\right)}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)} and \frac{x^{2}+400}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(\left(-x+60\right)^{2}+1600\right)-\left(x^{2}+400\right)}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)}$$

Do the multiplications in 2\left(\left(-x+60\right)^{2}+1600\right)-\left(x^{2}+400\right).

$$\frac{2x^{2}-240x+7200+3200-x^{2}-400}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)}$$

Combine like terms in 2x^{2}-240x+7200+3200-x^{2}-400.

$$\frac{x^{2}-240x+10000}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)}$$

Expand \left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right).

$$\frac{x^{2}-240x+10000}{x^{4}-120x^{3}+5600x^{2}-48000x+2080000}$$

Expand

$$\frac{x^{2}-240x+10000}{\left(x^{2}+400\right)\left(x^{2}-120x+5200\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}+400 and \left(60-x\right)^{2}+1600 is \left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right). Multiply \frac{2}{x^{2}+400} times \frac{\left(-x+60\right)^{2}+1600}{\left(-x+60\right)^{2}+1600}. Multiply \frac{1}{\left(60-x\right)^{2}+1600} times \frac{x^{2}+400}{x^{2}+400}.

$$\frac{2\left(\left(-x+60\right)^{2}+1600\right)}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)}-\frac{x^{2}+400}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)}$$

Since \frac{2\left(\left(-x+60\right)^{2}+1600\right)}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)} and \frac{x^{2}+400}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(\left(-x+60\right)^{2}+1600\right)-\left(x^{2}+400\right)}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)}$$

Do the multiplications in 2\left(\left(-x+60\right)^{2}+1600\right)-\left(x^{2}+400\right).

$$\frac{2x^{2}-240x+7200+3200-x^{2}-400}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)}$$

Combine like terms in 2x^{2}-240x+7200+3200-x^{2}-400.

$$\frac{x^{2}-240x+10000}{\left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right)}$$

Expand \left(x^{2}+400\right)\left(\left(-x+60\right)^{2}+1600\right).

$$\frac{x^{2}-240x+10000}{x^{4}-120x^{3}+5600x^{2}-48000x+2080000}$$