Solve (3/r+1/2)(r^2-1/4)

$$( \frac { 3 } { r } + \frac { 1 } { 2 } ) ( r ^ { 2 } - \frac { 1 } { 4 } )$$

Evaluate

$$\frac{\left(r+6\right)\left(4r^{2}-1\right)}{8r}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r and 2 is 2r. Multiply \frac{3}{r} times \frac{2}{2}. Multiply \frac{1}{2} times \frac{r}{r}.

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

Since \frac{3\times 2}{2r} and \frac{r}{2r} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\times 2+r.

$$\frac{6+r}{2r}\left(r^{2}-\frac{1}{4}\right)$$

Use the distributive property to multiply \frac{6+r}{2r} by r^{2}-\frac{1}{4}.

$$\frac{6+r}{2r}r^{2}-\frac{1}{4}\times \frac{6+r}{2r}$$

Express \frac{6+r}{2r}r^{2} as a single fraction.

$$\frac{\left(6+r\right)r^{2}}{2r}-\frac{1}{4}\times \frac{6+r}{2r}$$

Cancel out r in both numerator and denominator.

$$\frac{r\left(r+6\right)}{2}-\frac{1}{4}\times \frac{6+r}{2r}$$

Multiply -\frac{1}{4} times \frac{6+r}{2r} by multiplying numerator times numerator and denominator times denominator.

$$\frac{r\left(r+6\right)}{2}+\frac{-\left(6+r\right)}{4\times 2r}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4\times 2r is 2\times 4r. Multiply \frac{r\left(r+6\right)}{2} times \frac{4r}{4r}.

$$\frac{r\left(r+6\right)\times 4r}{2\times 4r}+\frac{-\left(6+r\right)}{2\times 4r}$$

Since \frac{r\left(r+6\right)\times 4r}{2\times 4r} and \frac{-\left(6+r\right)}{2\times 4r} have the same denominator, add them by adding their numerators.

$$\frac{r\left(r+6\right)\times 4r-\left(6+r\right)}{2\times 4r}$$

Do the multiplications in r\left(r+6\right)\times 4r-\left(6+r\right).

$$\frac{4r^{3}+24r^{2}-6-r}{2\times 4r}$$

Expand 2\times 4r.

$$\frac{4r^{3}+24r^{2}-6-r}{8r}$$

Expand

$$\frac{r^{2}}{2}+3r-\frac{1}{8}-\frac{3}{4r}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r and 2 is 2r. Multiply \frac{3}{r} times \frac{2}{2}. Multiply \frac{1}{2} times \frac{r}{r}.

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

Since \frac{3\times 2}{2r} and \frac{r}{2r} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\times 2+r.

$$\frac{6+r}{2r}\left(r^{2}-\frac{1}{4}\right)$$

Use the distributive property to multiply \frac{6+r}{2r} by r^{2}-\frac{1}{4}.

$$\frac{6+r}{2r}r^{2}-\frac{1}{4}\times \frac{6+r}{2r}$$

Express \frac{6+r}{2r}r^{2} as a single fraction.

$$\frac{\left(6+r\right)r^{2}}{2r}-\frac{1}{4}\times \frac{6+r}{2r}$$

Cancel out r in both numerator and denominator.

$$\frac{r\left(r+6\right)}{2}-\frac{1}{4}\times \frac{6+r}{2r}$$

Multiply -\frac{1}{4} times \frac{6+r}{2r} by multiplying numerator times numerator and denominator times denominator.

$$\frac{r\left(r+6\right)}{2}+\frac{-\left(6+r\right)}{4\times 2r}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4\times 2r is 2\times 4r. Multiply \frac{r\left(r+6\right)}{2} times \frac{4r}{4r}.

$$\frac{r\left(r+6\right)\times 4r}{2\times 4r}+\frac{-\left(6+r\right)}{2\times 4r}$$

Since \frac{r\left(r+6\right)\times 4r}{2\times 4r} and \frac{-\left(6+r\right)}{2\times 4r} have the same denominator, add them by adding their numerators.

$$\frac{r\left(r+6\right)\times 4r-\left(6+r\right)}{2\times 4r}$$

Do the multiplications in r\left(r+6\right)\times 4r-\left(6+r\right).

$$\frac{4r^{3}+24r^{2}-6-r}{2\times 4r}$$

Expand 2\times 4r.

$$\frac{4r^{3}+24r^{2}-6-r}{8r}$$