Solve 1/15=1/u+1/(80-u)

$$\frac { 1 } { 15 } = \frac { 1 } { u } + \frac { 1 } { ( 80 - u ) }$$

Solve for u

$$u=20$$

$$u=60$$

Steps Using the Quadratic Formula

Variable u cannot be equal to any of the values 0,80 since division by zero is not defined. Multiply both sides of the equation by 15u\left(u-80\right), the least common multiple of 15,u,80-u.

$$u\left(u-80\right)=15u-1200-15u$$

Use the distributive property to multiply u by u-80.

$$u^{2}-80u=15u-1200-15u$$

Combine 15u and -15u to get 0.

$$u^{2}-80u=-1200$$

Add 1200 to both sides.

$$u^{2}-80u+1200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -80 for b, and 1200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 1200}}{2}$$

Square -80.

$$u=\frac{-\left(-80\right)±\sqrt{6400-4\times 1200}}{2}$$

Multiply -4 times 1200.

$$u=\frac{-\left(-80\right)±\sqrt{6400-4800}}{2}$$

Add 6400 to -4800.

$$u=\frac{-\left(-80\right)±\sqrt{1600}}{2}$$

Take the square root of 1600.

$$u=\frac{-\left(-80\right)±40}{2}$$

The opposite of -80 is 80.

$$u=\frac{80±40}{2}$$

Now solve the equation u=\frac{80±40}{2} when ± is plus. Add 80 to 40.

$$u=\frac{120}{2}$$

Divide 120 by 2.

$$u=60$$

Now solve the equation u=\frac{80±40}{2} when ± is minus. Subtract 40 from 80.

$$u=\frac{40}{2}$$

Divide 40 by 2.

$$u=20$$

The equation is now solved.

$$u=60 u=20$$

Steps for Completing the Square

Variable u cannot be equal to any of the values 0,80 since division by zero is not defined. Multiply both sides of the equation by 15u\left(u-80\right), the least common multiple of 15,u,80-u.

$$u\left(u-80\right)=15u-1200-15u$$

Use the distributive property to multiply u by u-80.

$$u^{2}-80u=15u-1200-15u$$

Combine 15u and -15u to get 0.

$$u^{2}-80u=-1200$$

Divide -80, the coefficient of the x term, by 2 to get -40. Then add the square of -40 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$u^{2}-80u+\left(-40\right)^{2}=-1200+\left(-40\right)^{2}$$

Square -40.

$$u^{2}-80u+1600=-1200+1600$$

Add -1200 to 1600.

$$u^{2}-80u+1600=400$$

Factor u^{2}-80u+1600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(u-40\right)^{2}=400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(u-40\right)^{2}}=\sqrt{400}$$

Simplify.

$$u-40=20 u-40=-20$$

Add 40 to both sides of the equation.

$$u=60 u=20$$