Solve 1/1+4x^2=a/b+x^2

$$\frac { 1 } { 1 + 4 x ^ { 2 } } = \frac { a } { b + x ^ { 2 } }$$

Solve for a

$$a=\frac{x^{2}+b}{4x^{2}+1}$$

$$b\neq -x^{2}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(4x^{2}+1\right)\left(x^{2}+b\right), the least common multiple of 1+4x^{2},b+x^{2}.

$$x^{2}+b=\left(4x^{2}+1\right)a$$

Use the distributive property to multiply 4x^{2}+1 by a.

$$x^{2}+b=4x^{2}a+a$$

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

$$4x^{2}a+a=x^{2}+b$$

Combine all terms containing a.

$$\left(4x^{2}+1\right)a=x^{2}+b$$

Divide both sides by 4x^{2}+1.

$$\frac{\left(4x^{2}+1\right)a}{4x^{2}+1}=\frac{x^{2}+b}{4x^{2}+1}$$

Dividing by 4x^{2}+1 undoes the multiplication by 4x^{2}+1.

$$a=\frac{x^{2}+b}{4x^{2}+1}$$

Solve for b

$$b=4ax^{2}-x^{2}+a$$

$$a\neq 0$$