Solve u=3/uu+1/4u+4200

$$u = \frac { 3 } { u } u + \frac { 1 } { 4 } u + 4200$$

Solve for u

$$u=5604$$

Steps Using Factoring

Express \frac{3}{u}u as a single fraction.

$$u=\frac{3u}{u}+\frac{1}{4}u+4200$$

Subtract \frac{3u}{u} from both sides.

$$u-\frac{3u}{u}=\frac{1}{4}u+4200$$

To add or subtract expressions, expand them to make their denominators the same. Multiply u times \frac{u}{u}.

$$\frac{uu}{u}-\frac{3u}{u}=\frac{1}{4}u+4200$$

Since \frac{uu}{u} and \frac{3u}{u} have the same denominator, subtract them by subtracting their numerators.

$$\frac{uu-3u}{u}=\frac{1}{4}u+4200$$

Do the multiplications in uu-3u.

$$\frac{u^{2}-3u}{u}=\frac{1}{4}u+4200$$

Subtract \frac{1}{4}u from both sides.

$$\frac{u^{2}-3u}{u}-\frac{1}{4}u=4200$$

Variable u cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4u, the least common multiple of u,4.

$$4\left(u^{2}-3u\right)-\frac{1}{4}u\times 4u=16800u$$

Reorder the terms.

$$4\left(u^{2}-3u\right)-\frac{1}{4}\times 4uu=16800u$$

Multiply u and u to get u^{2}.

$$4\left(u^{2}-3u\right)-\frac{1}{4}\times 4u^{2}=16800u$$

Use the distributive property to multiply 4 by u^{2}-3u.

$$4u^{2}-12u-\frac{1}{4}\times 4u^{2}=16800u$$

Cancel out 4 and 4.

$$4u^{2}-12u-u^{2}=16800u$$

Combine 4u^{2} and -u^{2} to get 3u^{2}.

$$3u^{2}-12u=16800u$$

Subtract 16800u from both sides.

$$3u^{2}-12u-16800u=0$$

Combine -12u and -16800u to get -16812u.

$$3u^{2}-16812u=0$$

Factor out u.

$$u\left(3u-16812\right)=0$$

To find equation solutions, solve u=0 and 3u-16812=0.

$$u=0 u=5604$$

Variable u cannot be equal to 0.

$$u=5604$$

Steps Using the Quadratic Formula

Express \frac{3}{u}u as a single fraction.

$$u=\frac{3u}{u}+\frac{1}{4}u+4200$$

Subtract \frac{3u}{u} from both sides.

$$u-\frac{3u}{u}=\frac{1}{4}u+4200$$

To add or subtract expressions, expand them to make their denominators the same. Multiply u times \frac{u}{u}.

$$\frac{uu}{u}-\frac{3u}{u}=\frac{1}{4}u+4200$$

Since \frac{uu}{u} and \frac{3u}{u} have the same denominator, subtract them by subtracting their numerators.

$$\frac{uu-3u}{u}=\frac{1}{4}u+4200$$

Do the multiplications in uu-3u.

$$\frac{u^{2}-3u}{u}=\frac{1}{4}u+4200$$

Subtract \frac{1}{4}u from both sides.

$$\frac{u^{2}-3u}{u}-\frac{1}{4}u=4200$$

Subtract 4200 from both sides.

$$\frac{u^{2}-3u}{u}-\frac{1}{4}u-4200=0$$

Variable u cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4u, the least common multiple of u,4.

$$4\left(u^{2}-3u\right)-\frac{1}{4}u\times 4u+4u\left(-4200\right)=0$$

Reorder the terms.

$$4\left(u^{2}-3u\right)-\frac{1}{4}\times 4uu-4200\times 4u=0$$

Multiply u and u to get u^{2}.

$$4\left(u^{2}-3u\right)-\frac{1}{4}\times 4u^{2}-4200\times 4u=0$$

Use the distributive property to multiply 4 by u^{2}-3u.

$$4u^{2}-12u-\frac{1}{4}\times 4u^{2}-4200\times 4u=0$$

Cancel out 4 and 4.

$$4u^{2}-12u-u^{2}-4200\times 4u=0$$

Combine 4u^{2} and -u^{2} to get 3u^{2}.

$$3u^{2}-12u-4200\times 4u=0$$

Multiply -4200 and 4 to get -16800.

$$3u^{2}-12u-16800u=0$$

Combine -12u and -16800u to get -16812u.

$$3u^{2}-16812u=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -16812 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-\left(-16812\right)±\sqrt{\left(-16812\right)^{2}}}{2\times 3}$$

Take the square root of \left(-16812\right)^{2}.

$$u=\frac{-\left(-16812\right)±16812}{2\times 3}$$

The opposite of -16812 is 16812.

$$u=\frac{16812±16812}{2\times 3}$$

Multiply 2 times 3.

$$u=\frac{16812±16812}{6}$$

Now solve the equation u=\frac{16812±16812}{6} when ± is plus. Add 16812 to 16812.

$$u=\frac{33624}{6}$$

Divide 33624 by 6.

$$u=5604$$

Now solve the equation u=\frac{16812±16812}{6} when ± is minus. Subtract 16812 from 16812.

$$u=\frac{0}{6}$$

Divide 0 by 6.

$$u=0$$

The equation is now solved.

$$u=5604 u=0$$

Variable u cannot be equal to 0.

$$u=5604$$

Steps for Completing the Square

Express \frac{3}{u}u as a single fraction.

$$u=\frac{3u}{u}+\frac{1}{4}u+4200$$

Subtract \frac{3u}{u} from both sides.

$$u-\frac{3u}{u}=\frac{1}{4}u+4200$$

To add or subtract expressions, expand them to make their denominators the same. Multiply u times \frac{u}{u}.

$$\frac{uu}{u}-\frac{3u}{u}=\frac{1}{4}u+4200$$

Since \frac{uu}{u} and \frac{3u}{u} have the same denominator, subtract them by subtracting their numerators.

$$\frac{uu-3u}{u}=\frac{1}{4}u+4200$$

Do the multiplications in uu-3u.

$$\frac{u^{2}-3u}{u}=\frac{1}{4}u+4200$$

Subtract \frac{1}{4}u from both sides.

$$\frac{u^{2}-3u}{u}-\frac{1}{4}u=4200$$

Variable u cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4u, the least common multiple of u,4.

$$4\left(u^{2}-3u\right)-\frac{1}{4}u\times 4u=16800u$$

Reorder the terms.

$$4\left(u^{2}-3u\right)-\frac{1}{4}\times 4uu=16800u$$

Multiply u and u to get u^{2}.

$$4\left(u^{2}-3u\right)-\frac{1}{4}\times 4u^{2}=16800u$$

Use the distributive property to multiply 4 by u^{2}-3u.

$$4u^{2}-12u-\frac{1}{4}\times 4u^{2}=16800u$$

Cancel out 4 and 4.

$$4u^{2}-12u-u^{2}=16800u$$

Combine 4u^{2} and -u^{2} to get 3u^{2}.

$$3u^{2}-12u=16800u$$

Subtract 16800u from both sides.

$$3u^{2}-12u-16800u=0$$

Combine -12u and -16800u to get -16812u.

$$3u^{2}-16812u=0$$

Divide both sides by 3.

$$\frac{3u^{2}-16812u}{3}=\frac{0}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$u^{2}+\left(-\frac{16812}{3}\right)u=\frac{0}{3}$$

Divide -16812 by 3.

$$u^{2}-5604u=\frac{0}{3}$$

Divide 0 by 3.

$$u^{2}-5604u=0$$

Divide -5604, the coefficient of the x term, by 2 to get -2802. Then add the square of -2802 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$u^{2}-5604u+\left(-2802\right)^{2}=\left(-2802\right)^{2}$$

Square -2802.

$$u^{2}-5604u+7851204=7851204$$

Factor u^{2}-5604u+7851204. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(u-2802\right)^{2}=7851204$$

Take the square root of both sides of the equation.

$$\sqrt{\left(u-2802\right)^{2}}=\sqrt{7851204}$$

Simplify.

$$u-2802=2802 u-2802=-2802$$

Add 2802 to both sides of the equation.

$$u=5604 u=0$$

Variable u cannot be equal to 0.

$$u=5604$$