Solve f(-1/2p)=4(-1/2p)-p(-1/2p)+1/2p-12

$$f ( - \frac { 1 } { 2 } p ) = 4 ( - \frac { 1 } { 2 } p ) - p ( - \frac { 1 } { 2 } p ) + \frac { 1 } { 2 } p - 12$$

Evaluate

$$\frac{p^{2}-3p-24}{2}$$

Solution Steps

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

$$4\left(-\frac{1}{2}\right)p-p^{2}\left(-\frac{1}{2}\right)+\frac{1}{2}p-12$$

Express 4\left(-\frac{1}{2}\right) as a single fraction.

$$\frac{4\left(-1\right)}{2}p-p^{2}\left(-\frac{1}{2}\right)+\frac{1}{2}p-12$$

Multiply 4 and -1 to get -4.

$$\frac{-4}{2}p-p^{2}\left(-\frac{1}{2}\right)+\frac{1}{2}p-12$$

Divide -4 by 2 to get -2.

$$-2p-p^{2}\left(-\frac{1}{2}\right)+\frac{1}{2}p-12$$

Multiply -1 and -\frac{1}{2} to get \frac{1}{2}.

$$-2p+\frac{1}{2}p^{2}+\frac{1}{2}p-12$$

Combine -2p and \frac{1}{2}p to get -\frac{3}{2}p.

$$-\frac{3}{2}p+\frac{1}{2}p^{2}-12$$

Factor

$$\frac{\left(p-\frac{3-\sqrt{105}}{2}\right)\left(p-\frac{\sqrt{105}+3}{2}\right)}{2}$$