Solve l(l+1)=j(j+1)+[(j+1/2)+1]-3/4

$$l ( l + 1 ) = j ( j + 1 ) + [ ( j + \frac { 1 } { 2 } ) + 1 ] - \frac { 3 } { 4 }$$

Evaluate

$$j^{2}+2j+\frac{3}{4}$$

Solution Steps

Use the distributive property to multiply j by j+1.

$$j^{2}+j+j+\frac{1}{2}+1-\frac{3}{4}$$

Combine j and j to get 2j.

$$j^{2}+2j+\frac{1}{2}+1-\frac{3}{4}$$

Convert 1 to fraction \frac{2}{2}.

$$j^{2}+2j+\frac{1}{2}+\frac{2}{2}-\frac{3}{4}$$

Since \frac{1}{2} and \frac{2}{2} have the same denominator, add them by adding their numerators.

$$j^{2}+2j+\frac{1+2}{2}-\frac{3}{4}$$

Add 1 and 2 to get 3.

$$j^{2}+2j+\frac{3}{2}-\frac{3}{4}$$

Least common multiple of 2 and 4 is 4. Convert \frac{3}{2} and \frac{3}{4} to fractions with denominator 4.

$$j^{2}+2j+\frac{6}{4}-\frac{3}{4}$$

Since \frac{6}{4} and \frac{3}{4} have the same denominator, subtract them by subtracting their numerators.

$$j^{2}+2j+\frac{6-3}{4}$$

Subtract 3 from 6 to get 3.

$$j^{2}+2j+\frac{3}{4}$$

Expand

$$j^{2}+2j+\frac{3}{4}$$

Solution Steps

Use the distributive property to multiply j by j+1.

$$j^{2}+j+j+\frac{1}{2}+1-\frac{3}{4}$$

Combine j and j to get 2j.

$$j^{2}+2j+\frac{1}{2}+1-\frac{3}{4}$$

Convert 1 to fraction \frac{2}{2}.

$$j^{2}+2j+\frac{1}{2}+\frac{2}{2}-\frac{3}{4}$$

Since \frac{1}{2} and \frac{2}{2} have the same denominator, add them by adding their numerators.

$$j^{2}+2j+\frac{1+2}{2}-\frac{3}{4}$$

Add 1 and 2 to get 3.

$$j^{2}+2j+\frac{3}{2}-\frac{3}{4}$$

Least common multiple of 2 and 4 is 4. Convert \frac{3}{2} and \frac{3}{4} to fractions with denominator 4.

$$j^{2}+2j+\frac{6}{4}-\frac{3}{4}$$

Since \frac{6}{4} and \frac{3}{4} have the same denominator, subtract them by subtracting their numerators.

$$j^{2}+2j+\frac{6-3}{4}$$

Subtract 3 from 6 to get 3.

$$j^{2}+2j+\frac{3}{4}$$