Solve yM=frac{(a^{2}+b^{2})^{2}-(a^{2}-b^2)^2}{ab[(a+b)^2+(a-b)^2]}

$$y M = \frac { ( a ^ { 2 } + b ^ { 2 } ) ^ { 2 } - ( a ^ { 2 } - b ^ { 2 } ) ^ { 2 } } { a b [ ( a + b ) ^ { 2 } + ( a - b ) ^ { 2 } ] }$$

Solve for M

$$M=\frac{2ab}{y\left(a^{2}+b^{2}\right)}$$

$$a\neq 0\text{ and }y\neq 0\text{ and }b\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2ab\left(a^{2}+b^{2}\right).

$$yM\times 2ab\left(a^{2}+b^{2}\right)=\left(a^{2}+b^{2}\right)^{2}-\left(a^{2}-b^{2}\right)^{2}$$

Use the distributive property to multiply yM\times 2ab by a^{2}+b^{2}.

$$2yMba^{3}+2yMab^{3}=\left(a^{2}+b^{2}\right)^{2}-\left(a^{2}-b^{2}\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a^{2}+b^{2}\right)^{2}.

$$2yMba^{3}+2yMab^{3}=\left(a^{2}\right)^{2}+2a^{2}b^{2}+\left(b^{2}\right)^{2}-\left(a^{2}-b^{2}\right)^{2}$$

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

$$2yMba^{3}+2yMab^{3}=a^{4}+2a^{2}b^{2}+\left(b^{2}\right)^{2}-\left(a^{2}-b^{2}\right)^{2}$$

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

$$2yMba^{3}+2yMab^{3}=a^{4}+2a^{2}b^{2}+b^{4}-\left(a^{2}-b^{2}\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a^{2}-b^{2}\right)^{2}.

$$2yMba^{3}+2yMab^{3}=a^{4}+2a^{2}b^{2}+b^{4}-\left(\left(a^{2}\right)^{2}-2a^{2}b^{2}+\left(b^{2}\right)^{2}\right)$$

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

$$2yMba^{3}+2yMab^{3}=a^{4}+2a^{2}b^{2}+b^{4}-\left(a^{4}-2a^{2}b^{2}+\left(b^{2}\right)^{2}\right)$$

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

$$2yMba^{3}+2yMab^{3}=a^{4}+2a^{2}b^{2}+b^{4}-\left(a^{4}-2a^{2}b^{2}+b^{4}\right)$$

To find the opposite of a^{4}-2a^{2}b^{2}+b^{4}, find the opposite of each term.

$$2yMba^{3}+2yMab^{3}=a^{4}+2a^{2}b^{2}+b^{4}-a^{4}+2a^{2}b^{2}-b^{4}$$

Combine a^{4} and -a^{4} to get 0.

$$2yMba^{3}+2yMab^{3}=2a^{2}b^{2}+b^{4}+2a^{2}b^{2}-b^{4}$$

Combine 2a^{2}b^{2} and 2a^{2}b^{2} to get 4a^{2}b^{2}.

$$2yMba^{3}+2yMab^{3}=4a^{2}b^{2}+b^{4}-b^{4}$$

Combine b^{4} and -b^{4} to get 0.

$$2yMba^{3}+2yMab^{3}=4a^{2}b^{2}$$

Combine all terms containing M.

$$\left(2yba^{3}+2yab^{3}\right)M=4a^{2}b^{2}$$

The equation is in standard form.

$$\left(2ayb^{3}+2bya^{3}\right)M=4a^{2}b^{2}$$

Divide both sides by 2yba^{3}+2yab^{3}.

$$\frac{\left(2ayb^{3}+2bya^{3}\right)M}{2ayb^{3}+2bya^{3}}=\frac{4a^{2}b^{2}}{2ayb^{3}+2bya^{3}}$$

Dividing by 2yba^{3}+2yab^{3} undoes the multiplication by 2yba^{3}+2yab^{3}.

$$M=\frac{4a^{2}b^{2}}{2ayb^{3}+2bya^{3}}$$

Divide 4a^{2}b^{2} by 2yba^{3}+2yab^{3}.

$$M=\frac{2ab}{y\left(a^{2}+b^{2}\right)}$$

Solve for a

$$a=\frac{b\left(\sqrt{1-\left(My\right)^{2}}+1\right)}{My}$$

$$a=\frac{b\left(-\sqrt{1-\left(My\right)^{2}}+1\right)}{My}\text{, }y\neq 0\text{ and }b\neq 0\text{ and }M\neq 0\text{ and }|y|\leq \frac{1}{|M|}$$