Solve 40(1/60-x)^2+20/x^2/1+(frac{40{60-x})^2}

$$\frac { 40 ( \frac { 1 } { 60 - x } ) ^ { 2 } + 20 / x ^ { 2 } } { 1 + ( \frac { 40 } { 60 - x } ) ^ { 2 } }$$

Evaluate

$$\frac{60\left(x^{2}-40x+1200\right)}{x^{2}\left(x^{2}-120x+5200\right)}$$

Short Solution Steps

To raise \frac{1}{60-x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{40\times \frac{1^{2}}{\left(60-x\right)^{2}}+\frac{20}{x^{2}}}{1+\left(\frac{40}{60-x}\right)^{2}}$$

Express 40\times \frac{1^{2}}{\left(60-x\right)^{2}} as a single fraction.

$$\frac{\frac{40\times 1^{2}}{\left(60-x\right)^{2}}+\frac{20}{x^{2}}}{1+\left(\frac{40}{60-x}\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(60-x\right)^{2} and x^{2} is x^{2}\left(-x+60\right)^{2}. Multiply \frac{40\times 1^{2}}{\left(60-x\right)^{2}} times \frac{x^{2}}{x^{2}}. Multiply \frac{20}{x^{2}} times \frac{\left(-x+60\right)^{2}}{\left(-x+60\right)^{2}}.

$$\frac{\frac{40\times 1^{2}x^{2}}{x^{2}\left(-x+60\right)^{2}}+\frac{20\left(-x+60\right)^{2}}{x^{2}\left(-x+60\right)^{2}}}{1+\left(\frac{40}{60-x}\right)^{2}}$$

Since \frac{40\times 1^{2}x^{2}}{x^{2}\left(-x+60\right)^{2}} and \frac{20\left(-x+60\right)^{2}}{x^{2}\left(-x+60\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{40\times 1^{2}x^{2}+20\left(-x+60\right)^{2}}{x^{2}\left(-x+60\right)^{2}}}{1+\left(\frac{40}{60-x}\right)^{2}}$$

Do the multiplications in 40\times 1^{2}x^{2}+20\left(-x+60\right)^{2}.

$$\frac{\frac{40x^{2}+20x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{1+\left(\frac{40}{60-x}\right)^{2}}$$

Combine like terms in 40x^{2}+20x^{2}-2400x+72000.

$$\frac{\frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{1+\left(\frac{40}{60-x}\right)^{2}}$$

To raise \frac{40}{60-x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{1+\frac{40^{2}}{\left(60-x\right)^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(60-x\right)^{2}}{\left(60-x\right)^{2}}.

$$\frac{\frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{\frac{\left(60-x\right)^{2}}{\left(60-x\right)^{2}}+\frac{40^{2}}{\left(60-x\right)^{2}}}$$

Since \frac{\left(60-x\right)^{2}}{\left(60-x\right)^{2}} and \frac{40^{2}}{\left(60-x\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{\frac{\left(60-x\right)^{2}+40^{2}}{\left(60-x\right)^{2}}}$$

Do the multiplications in \left(60-x\right)^{2}+40^{2}.

$$\frac{\frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{\frac{3600-120x+x^{2}+40^{2}}{\left(60-x\right)^{2}}}$$

Combine like terms in 3600-120x+x^{2}+40^{2}.

$$\frac{\frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{\frac{5200-120x+x^{2}}{\left(60-x\right)^{2}}}$$

Divide \frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}} by \frac{5200-120x+x^{2}}{\left(60-x\right)^{2}} by multiplying \frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}} by the reciprocal of \frac{5200-120x+x^{2}}{\left(60-x\right)^{2}}.

$$\frac{\left(60x^{2}-2400x+72000\right)\left(60-x\right)^{2}}{x^{2}\left(-x+60\right)^{2}\left(5200-120x+x^{2}\right)}$$

Cancel out \left(-x+60\right)^{2} in both numerator and denominator.

$$\frac{60x^{2}-2400x+72000}{x^{2}\left(x^{2}-120x+5200\right)}$$

Use the distributive property to multiply x^{2} by x^{2}-120x+5200.

$$\frac{60x^{2}-2400x+72000}{x^{4}-120x^{3}+5200x^{2}}$$

Expand

$$\frac{60\left(x^{2}-40x+1200\right)}{x^{2}\left(x^{2}-120x+5200\right)}$$

Short Solution Steps

To raise \frac{1}{60-x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{40\times \frac{1^{2}}{\left(60-x\right)^{2}}+\frac{20}{x^{2}}}{1+\left(\frac{40}{60-x}\right)^{2}}$$

Express 40\times \frac{1^{2}}{\left(60-x\right)^{2}} as a single fraction.

$$\frac{\frac{40\times 1^{2}}{\left(60-x\right)^{2}}+\frac{20}{x^{2}}}{1+\left(\frac{40}{60-x}\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(60-x\right)^{2} and x^{2} is x^{2}\left(-x+60\right)^{2}. Multiply \frac{40\times 1^{2}}{\left(60-x\right)^{2}} times \frac{x^{2}}{x^{2}}. Multiply \frac{20}{x^{2}} times \frac{\left(-x+60\right)^{2}}{\left(-x+60\right)^{2}}.

$$\frac{\frac{40\times 1^{2}x^{2}}{x^{2}\left(-x+60\right)^{2}}+\frac{20\left(-x+60\right)^{2}}{x^{2}\left(-x+60\right)^{2}}}{1+\left(\frac{40}{60-x}\right)^{2}}$$

Since \frac{40\times 1^{2}x^{2}}{x^{2}\left(-x+60\right)^{2}} and \frac{20\left(-x+60\right)^{2}}{x^{2}\left(-x+60\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{40\times 1^{2}x^{2}+20\left(-x+60\right)^{2}}{x^{2}\left(-x+60\right)^{2}}}{1+\left(\frac{40}{60-x}\right)^{2}}$$

Do the multiplications in 40\times 1^{2}x^{2}+20\left(-x+60\right)^{2}.

$$\frac{\frac{40x^{2}+20x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{1+\left(\frac{40}{60-x}\right)^{2}}$$

Combine like terms in 40x^{2}+20x^{2}-2400x+72000.

$$\frac{\frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{1+\left(\frac{40}{60-x}\right)^{2}}$$

To raise \frac{40}{60-x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{1+\frac{40^{2}}{\left(60-x\right)^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(60-x\right)^{2}}{\left(60-x\right)^{2}}.

$$\frac{\frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{\frac{\left(60-x\right)^{2}}{\left(60-x\right)^{2}}+\frac{40^{2}}{\left(60-x\right)^{2}}}$$

Since \frac{\left(60-x\right)^{2}}{\left(60-x\right)^{2}} and \frac{40^{2}}{\left(60-x\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{\frac{\left(60-x\right)^{2}+40^{2}}{\left(60-x\right)^{2}}}$$

Do the multiplications in \left(60-x\right)^{2}+40^{2}.

$$\frac{\frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{\frac{3600-120x+x^{2}+40^{2}}{\left(60-x\right)^{2}}}$$

Combine like terms in 3600-120x+x^{2}+40^{2}.

$$\frac{\frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}}}{\frac{5200-120x+x^{2}}{\left(60-x\right)^{2}}}$$

Divide \frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}} by \frac{5200-120x+x^{2}}{\left(60-x\right)^{2}} by multiplying \frac{60x^{2}-2400x+72000}{x^{2}\left(-x+60\right)^{2}} by the reciprocal of \frac{5200-120x+x^{2}}{\left(60-x\right)^{2}}.

$$\frac{\left(60x^{2}-2400x+72000\right)\left(60-x\right)^{2}}{x^{2}\left(-x+60\right)^{2}\left(5200-120x+x^{2}\right)}$$

Cancel out \left(-x+60\right)^{2} in both numerator and denominator.

$$\frac{60x^{2}-2400x+72000}{x^{2}\left(x^{2}-120x+5200\right)}$$

Use the distributive property to multiply x^{2} by x^{2}-120x+5200.

$$\frac{60x^{2}-2400x+72000}{x^{4}-120x^{3}+5200x^{2}}$$