Solve k*(-1/2)^3-(k+2)(-1/2)^2+lambda(-1/2)+L=f

$$k \cdot ( - \frac { 1 } { 2 } ) ^ { 3 } - ( k + 2 ) ( - \frac { 1 } { 2 } ) ^ { 2 } + \lambda ( - \frac { 1 } { 2 } ) + L = f$$

Solve for L

$$L=\frac{\lambda }{2}+\frac{3k}{8}+f+\frac{1}{2}$$

Solution Steps

Calculate -\frac{1}{2} to the power of 3 and get -\frac{1}{8}.

$$k\left(-\frac{1}{8}\right)-\left(k+2\right)\left(-\frac{1}{2}\right)^{2}+\lambda \left(-\frac{1}{2}\right)+L=f$$

Calculate -\frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$k\left(-\frac{1}{8}\right)-\left(k+2\right)\times \frac{1}{4}+\lambda \left(-\frac{1}{2}\right)+L=f$$

Use the distributive property to multiply k+2 by \frac{1}{4}.

$$k\left(-\frac{1}{8}\right)-\left(\frac{1}{4}k+\frac{1}{2}\right)+\lambda \left(-\frac{1}{2}\right)+L=f$$

To find the opposite of \frac{1}{4}k+\frac{1}{2}, find the opposite of each term.

$$k\left(-\frac{1}{8}\right)-\frac{1}{4}k-\frac{1}{2}+\lambda \left(-\frac{1}{2}\right)+L=f$$

Combine k\left(-\frac{1}{8}\right) and -\frac{1}{4}k to get -\frac{3}{8}k.

$$-\frac{3}{8}k-\frac{1}{2}+\lambda \left(-\frac{1}{2}\right)+L=f$$

Add \frac{3}{8}k to both sides.

$$-\frac{1}{2}+\lambda \left(-\frac{1}{2}\right)+L=f+\frac{3}{8}k$$

Add \frac{1}{2} to both sides.

$$\lambda \left(-\frac{1}{2}\right)+L=f+\frac{3}{8}k+\frac{1}{2}$$

Subtract \lambda \left(-\frac{1}{2}\right) from both sides.

$$L=f+\frac{3}{8}k+\frac{1}{2}-\lambda \left(-\frac{1}{2}\right)$$

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

$$L=f+\frac{3}{8}k+\frac{1}{2}+\frac{1}{2}\lambda$$

Solve for f

$$f=-\frac{\lambda }{2}-\frac{3k}{8}+L-\frac{1}{2}$$