Solve t=1/(-frac{1{2r^2}+2/b^2)}dr

$$t = \frac { 1 } { ( - \frac { 1 } { 2 r ^ { 2 } } + \frac { 2 } { b ^ { 2 } } ) } d r$$

Solve for d (complex solution)

$$d=-\frac{t}{2r^{3}}+\frac{2t}{rb^{2}}$$

$$r\neq 0\text{ and }b\neq 0\text{ and }r\neq \frac{b}{2}\text{ and }r\neq -\frac{b}{2}$$

Steps for Solving Linear Equation

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

$$t=\frac{1}{-\frac{b^{2}}{2b^{2}r^{2}}+\frac{2\times 2r^{2}}{2b^{2}r^{2}}}dr$$

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

$$t=\frac{1}{\frac{-b^{2}+2\times 2r^{2}}{2b^{2}r^{2}}}dr$$

Do the multiplications in -b^{2}+2\times 2r^{2}.

$$t=\frac{1}{\frac{-b^{2}+4r^{2}}{2b^{2}r^{2}}}dr$$

Divide 1 by \frac{-b^{2}+4r^{2}}{2b^{2}r^{2}} by multiplying 1 by the reciprocal of \frac{-b^{2}+4r^{2}}{2b^{2}r^{2}}.

$$t=\frac{2b^{2}r^{2}}{-b^{2}+4r^{2}}dr$$

Express \frac{2b^{2}r^{2}}{-b^{2}+4r^{2}}d as a single fraction.

$$t=\frac{2b^{2}r^{2}d}{-b^{2}+4r^{2}}r$$

Express \frac{2b^{2}r^{2}d}{-b^{2}+4r^{2}}r as a single fraction.

$$t=\frac{2b^{2}r^{2}dr}{-b^{2}+4r^{2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$t=\frac{2b^{2}r^{3}d}{-b^{2}+4r^{2}}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{2b^{2}r^{3}d}{-b^{2}+4r^{2}}=t$$

Multiply both sides of the equation by \left(-2r+b\right)\left(-2r-b\right).

$$2b^{2}r^{3}d=t\left(-2r+b\right)\left(-2r-b\right)$$

Reorder the terms.

$$2db^{2}r^{3}=t\left(-2r+b\right)\left(-2r-b\right)$$

Use the distributive property to multiply t by -2r+b.

$$2db^{2}r^{3}=\left(-2tr+tb\right)\left(-2r-b\right)$$

Use the distributive property to multiply -2tr+tb by -2r-b and combine like terms.

$$2db^{2}r^{3}=4tr^{2}-tb^{2}$$

The equation is in standard form.

$$2b^{2}r^{3}d=4tr^{2}-tb^{2}$$

Divide both sides by 2b^{2}r^{3}.

$$\frac{2b^{2}r^{3}d}{2b^{2}r^{3}}=\frac{t\left(2r-b\right)\left(2r+b\right)}{2b^{2}r^{3}}$$

Dividing by 2b^{2}r^{3} undoes the multiplication by 2b^{2}r^{3}.

$$d=\frac{t\left(2r-b\right)\left(2r+b\right)}{2b^{2}r^{3}}$$

Divide t\left(2r-b\right)\left(2r+b\right) by 2b^{2}r^{3}.

$$d=-\frac{t}{2r^{3}}+\frac{2t}{rb^{2}}$$

Solve for d

$$d=-\frac{t}{2r^{3}}+\frac{2t}{rb^{2}}$$

$$r\neq 0\text{ and }b\neq 0\text{ and }|r|\neq \frac{|b|}{2}$$

Steps for Solving Linear Equation

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

$$t=\frac{1}{-\frac{b^{2}}{2b^{2}r^{2}}+\frac{2\times 2r^{2}}{2b^{2}r^{2}}}dr$$

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

$$t=\frac{1}{\frac{-b^{2}+2\times 2r^{2}}{2b^{2}r^{2}}}dr$$

Do the multiplications in -b^{2}+2\times 2r^{2}.

$$t=\frac{1}{\frac{-b^{2}+4r^{2}}{2b^{2}r^{2}}}dr$$

Divide 1 by \frac{-b^{2}+4r^{2}}{2b^{2}r^{2}} by multiplying 1 by the reciprocal of \frac{-b^{2}+4r^{2}}{2b^{2}r^{2}}.

$$t=\frac{2b^{2}r^{2}}{-b^{2}+4r^{2}}dr$$

Express \frac{2b^{2}r^{2}}{-b^{2}+4r^{2}}d as a single fraction.

$$t=\frac{2b^{2}r^{2}d}{-b^{2}+4r^{2}}r$$

Express \frac{2b^{2}r^{2}d}{-b^{2}+4r^{2}}r as a single fraction.

$$t=\frac{2b^{2}r^{2}dr}{-b^{2}+4r^{2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$t=\frac{2b^{2}r^{3}d}{-b^{2}+4r^{2}}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{2b^{2}r^{3}d}{-b^{2}+4r^{2}}=t$$

Multiply both sides of the equation by \left(-2r+b\right)\left(-2r-b\right).

$$2b^{2}r^{3}d=t\left(-2r+b\right)\left(-2r-b\right)$$

Reorder the terms.

$$2db^{2}r^{3}=t\left(-2r+b\right)\left(-2r-b\right)$$

Use the distributive property to multiply t by -2r+b.

$$2db^{2}r^{3}=\left(-2tr+tb\right)\left(-2r-b\right)$$

Use the distributive property to multiply -2tr+tb by -2r-b and combine like terms.

$$2db^{2}r^{3}=4tr^{2}-tb^{2}$$

The equation is in standard form.

$$2b^{2}r^{3}d=4tr^{2}-tb^{2}$$

Divide both sides by 2b^{2}r^{3}.

$$\frac{2b^{2}r^{3}d}{2b^{2}r^{3}}=\frac{t\left(2r-b\right)\left(2r+b\right)}{2b^{2}r^{3}}$$

Dividing by 2b^{2}r^{3} undoes the multiplication by 2b^{2}r^{3}.

$$d=\frac{t\left(2r-b\right)\left(2r+b\right)}{2b^{2}r^{3}}$$

Divide t\left(2r-b\right)\left(2r+b\right) by 2b^{2}r^{3}.

$$d=-\frac{t}{2r^{3}}+\frac{2t}{rb^{2}}$$