Solve c^2-4d^2/3L+32/3=0

$$c ^ { 2 } - \frac { 4 d ^ { 2 } } { 3 } L + \frac { 32 } { 3 } = 0$$

Solve for L (complex solution)

$$\left\{\begin{matrix}L=\frac{3c^{2}+32}{4d^{2}}\text{, }&d\neq 0\\L\in \mathrm{C}\text{, }&\left(c=\frac{4\sqrt{6}i}{3}\text{ or }c=-\frac{4\sqrt{6}i}{3}\right)\text{ and }d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 3.

$$3\left(c^{2}-\frac{4d^{2}}{3}L\right)+32=0$$

Express \frac{4d^{2}}{3}L as a single fraction.

$$3\left(c^{2}-\frac{4d^{2}L}{3}\right)+32=0$$

Use the distributive property to multiply 3 by c^{2}-\frac{4d^{2}L}{3}.

$$3c^{2}+3\left(-\frac{4d^{2}L}{3}\right)+32=0$$

Express 3\left(-\frac{4d^{2}L}{3}\right) as a single fraction.

$$3c^{2}+\frac{-3\times 4d^{2}L}{3}+32=0$$

Cancel out 3 and 3.

$$3c^{2}-4d^{2}L+32=0$$

Subtract 3c^{2} from both sides. Anything subtracted from zero gives its negation.

$$-4d^{2}L+32=-3c^{2}$$

Subtract 32 from both sides.

$$-4d^{2}L=-3c^{2}-32$$

The equation is in standard form.

$$\left(-4d^{2}\right)L=-3c^{2}-32$$

Divide both sides by -4d^{2}.

$$\frac{\left(-4d^{2}\right)L}{-4d^{2}}=\frac{-3c^{2}-32}{-4d^{2}}$$

Dividing by -4d^{2} undoes the multiplication by -4d^{2}.

$$L=\frac{-3c^{2}-32}{-4d^{2}}$$

Divide -3c^{2}-32 by -4d^{2}.

$$L=\frac{3c^{2}+32}{4d^{2}}$$

Solve for L

$$L=\frac{3c^{2}+32}{4d^{2}}$$

$$d\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 3.

$$3\left(c^{2}-\frac{4d^{2}}{3}L\right)+32=0$$

Express \frac{4d^{2}}{3}L as a single fraction.

$$3\left(c^{2}-\frac{4d^{2}L}{3}\right)+32=0$$

Use the distributive property to multiply 3 by c^{2}-\frac{4d^{2}L}{3}.

$$3c^{2}+3\left(-\frac{4d^{2}L}{3}\right)+32=0$$

Express 3\left(-\frac{4d^{2}L}{3}\right) as a single fraction.

$$3c^{2}+\frac{-3\times 4d^{2}L}{3}+32=0$$

Cancel out 3 and 3.

$$3c^{2}-4d^{2}L+32=0$$

Subtract 3c^{2} from both sides. Anything subtracted from zero gives its negation.

$$-4d^{2}L+32=-3c^{2}$$

Subtract 32 from both sides.

$$-4d^{2}L=-3c^{2}-32$$

The equation is in standard form.

$$\left(-4d^{2}\right)L=-3c^{2}-32$$

Divide both sides by -4d^{2}.

$$\frac{\left(-4d^{2}\right)L}{-4d^{2}}=\frac{-3c^{2}-32}{-4d^{2}}$$

Dividing by -4d^{2} undoes the multiplication by -4d^{2}.

$$L=\frac{-3c^{2}-32}{-4d^{2}}$$

Divide -3c^{2}-32 by -4d^{2}.

$$L=\frac{3c^{2}+32}{4d^{2}}$$

Solve for c (complex solution)

$$c=-\frac{2\sqrt{3\left(Ld^{2}-8\right)}}{3}$$

$$c=\frac{2\sqrt{3\left(Ld^{2}-8\right)}}{3}$$

Solve for c

$$c=\frac{2\sqrt{3\left(Ld^{2}-8\right)}}{3}$$

$$c=-\frac{2\sqrt{3\left(Ld^{2}-8\right)}}{3}\text{, }L\geq \frac{8}{d^{2}}\text{ or }d=0$$