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

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

Evaluate

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

Solution Steps

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

$$j^{2}+j-\left(-j-\frac{1}{2}\right)-\frac{3}{4}$$

To find the opposite of -j-\frac{1}{2}, find the opposite of each term.

$$j^{2}+j-\left(-j\right)-\left(-\frac{1}{2}\right)-\frac{3}{4}$$

The opposite of -\frac{1}{2} is \frac{1}{2}.

$$j^{2}+j-\left(-j\right)+\frac{1}{2}-\frac{3}{4}$$

Least common multiple of 2 and 4 is 4. Convert \frac{1}{2} and \frac{3}{4} to fractions with denominator 4.

$$j^{2}+j-\left(-j\right)+\frac{2}{4}-\frac{3}{4}$$

Since \frac{2}{4} and \frac{3}{4} have the same denominator, subtract them by subtracting their numerators.

$$j^{2}+j-\left(-j\right)+\frac{2-3}{4}$$

Subtract 3 from 2 to get -1.

$$j^{2}+j-\left(-j\right)-\frac{1}{4}$$

Multiply -1 and -1 to get 1.

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

Combine j and j to get 2j.

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

Expand

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

Solution Steps

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

$$j^{2}+j-\left(-j-\frac{1}{2}\right)-\frac{3}{4}$$

To find the opposite of -j-\frac{1}{2}, find the opposite of each term.

$$j^{2}+j-\left(-j\right)-\left(-\frac{1}{2}\right)-\frac{3}{4}$$

The opposite of -\frac{1}{2} is \frac{1}{2}.

$$j^{2}+j-\left(-j\right)+\frac{1}{2}-\frac{3}{4}$$

Least common multiple of 2 and 4 is 4. Convert \frac{1}{2} and \frac{3}{4} to fractions with denominator 4.

$$j^{2}+j-\left(-j\right)+\frac{2}{4}-\frac{3}{4}$$

Since \frac{2}{4} and \frac{3}{4} have the same denominator, subtract them by subtracting their numerators.

$$j^{2}+j-\left(-j\right)+\frac{2-3}{4}$$

Subtract 3 from 2 to get -1.

$$j^{2}+j-\left(-j\right)-\frac{1}{4}$$

Multiply -1 and -1 to get 1.

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

Combine j and j to get 2j.

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