Solve (-3/5t+2)^2+(4/5t^2)^2=4/5q

$$( - \frac { 3 } { 5 } t + 2 ) ^ { 2 } + ( \frac { 4 } { 5 } t ^ { 2 } ) ^ { 2 } = \frac { 4 } { 5 q }$$

Solve for q

$$q=-\frac{20}{-16t^{4}-9t^{2}+60t-100}$$

$$-16t^{4}-\left(3t-10\right)^{2}\neq 0$$

Steps for Solving Linear Equation

Variable q cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 5q.

$$5q\left(-\frac{3}{5}t+2\right)^{2}+5q\times \left(\frac{4}{5}t^{2}\right)^{2}=4$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-\frac{3}{5}t+2\right)^{2}.

$$5q\left(\frac{9}{25}t^{2}-\frac{12}{5}t+4\right)+5q\times \left(\frac{4}{5}t^{2}\right)^{2}=4$$

Use the distributive property to multiply 5q by \frac{9}{25}t^{2}-\frac{12}{5}t+4.

$$\frac{9}{5}t^{2}q-12tq+20q+5q\times \left(\frac{4}{5}t^{2}\right)^{2}=4$$

Expand \left(\frac{4}{5}t^{2}\right)^{2}.

$$\frac{9}{5}t^{2}q-12tq+20q+5q\times \left(\frac{4}{5}\right)^{2}\left(t^{2}\right)^{2}=4$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{9}{5}t^{2}q-12tq+20q+5q\times \left(\frac{4}{5}\right)^{2}t^{4}=4$$

Calculate \frac{4}{5} to the power of 2 and get \frac{16}{25}.

$$\frac{9}{5}t^{2}q-12tq+20q+5q\times \frac{16}{25}t^{4}=4$$

Multiply 5 and \frac{16}{25} to get \frac{16}{5}.

$$\frac{9}{5}t^{2}q-12tq+20q+\frac{16}{5}qt^{4}=4$$

Combine all terms containing q.

$$\left(\frac{9}{5}t^{2}-12t+20+\frac{16}{5}t^{4}\right)q=4$$

The equation is in standard form.

$$\left(\frac{16t^{4}}{5}+\frac{9t^{2}}{5}-12t+20\right)q=4$$

Divide both sides by \frac{9}{5}t^{2}-12t+20+\frac{16}{5}t^{4}.

$$\frac{\left(\frac{16t^{4}}{5}+\frac{9t^{2}}{5}-12t+20\right)q}{\frac{16t^{4}}{5}+\frac{9t^{2}}{5}-12t+20}=\frac{4}{\frac{16t^{4}}{5}+\frac{9t^{2}}{5}-12t+20}$$

Dividing by \frac{9}{5}t^{2}-12t+20+\frac{16}{5}t^{4} undoes the multiplication by \frac{9}{5}t^{2}-12t+20+\frac{16}{5}t^{4}.

$$q=\frac{4}{\frac{16t^{4}}{5}+\frac{9t^{2}}{5}-12t+20}$$

Divide 4 by \frac{9}{5}t^{2}-12t+20+\frac{16}{5}t^{4}.

$$q=\frac{20}{16t^{4}+9t^{2}-60t+100}$$

Variable q cannot be equal to 0.

$$q=\frac{20}{16t^{4}+9t^{2}-60t+100}\text{, }q\neq 0$$