Solve a+1/beta^2+1=a/beta

$$\frac { a + 1 } { \beta ^ { 2 } + 1 } = \frac { a } { \beta }$$

Solve for a

$$a=-\frac{\beta }{-\beta ^{2}+\beta -1}$$

$$\beta \neq 0$$

Steps for Solving Linear Equation

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

$$\beta \left(a+1\right)=\left(\beta ^{2}+1\right)a$$

Use the distributive property to multiply \beta by a+1.

$$\beta a+\beta =\left(\beta ^{2}+1\right)a$$

Use the distributive property to multiply \beta ^{2}+1 by a.

$$\beta a+\beta =\beta ^{2}a+a$$

Subtract \beta ^{2}a from both sides.

$$\beta a+\beta -\beta ^{2}a=a$$

Subtract a from both sides.

$$\beta a+\beta -\beta ^{2}a-a=0$$

Subtract \beta from both sides. Anything subtracted from zero gives its negation.

$$\beta a-\beta ^{2}a-a=-\beta$$

Combine all terms containing a.

$$\left(\beta -\beta ^{2}-1\right)a=-\beta$$

The equation is in standard form.

$$\left(-\beta ^{2}+\beta -1\right)a=-\beta$$

Divide both sides by -\beta ^{2}+\beta -1.

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

Dividing by -\beta ^{2}+\beta -1 undoes the multiplication by -\beta ^{2}+\beta -1.

$$a=-\frac{\beta }{-\beta ^{2}+\beta -1}$$

Solve for β

$$\beta =-\frac{\sqrt{\left(1-a\right)\left(3a+1\right)}-a-1}{2a}$$

$$\beta =\frac{\sqrt{\left(1-a\right)\left(3a+1\right)}+a+1}{2a}\text{, }a\neq 0\text{ and }a\geq -\frac{1}{3}\text{ and }a\leq 1$$