Solve a-1/a^2+a-1/2a-2-1/2a+2=1-2a/a(a^2-1)

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

Solve for a (complex solution)

$$a\in \mathrm{C}\setminus -1,0,1$$

Solution Steps

Variable a cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by 2a\left(a-1\right)\left(a+1\right), the least common multiple of a^{2}+a,2a-2,2a+2,a\left(a^{2}-1\right).

$$\left(2a-2\right)\left(a-1\right)-\left(a+1\right)a-\left(a-1\right)a=2\left(1-2a\right)$$

Use the distributive property to multiply 2a-2 by a-1 and combine like terms.

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

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

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

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

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

Combine 2a^{2} and -a^{2} to get a^{2}.

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

Combine -4a and -a to get -5a.

$$a^{2}-5a+2-\left(a-1\right)a=2\left(1-2a\right)$$

Use the distributive property to multiply a-1 by a.

$$a^{2}-5a+2-\left(a^{2}-a\right)=2\left(1-2a\right)$$

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

$$a^{2}-5a+2-a^{2}+a=2\left(1-2a\right)$$

Combine a^{2} and -a^{2} to get 0.

$$-5a+2+a=2\left(1-2a\right)$$

Combine -5a and a to get -4a.

$$-4a+2=2\left(1-2a\right)$$

Use the distributive property to multiply 2 by 1-2a.

$$-4a+2=2-4a$$

Subtract 2 from both sides.

$$-4a+2-2=-4a$$

Subtract 2 from 2 to get 0.

$$-4a=-4a$$

Add 4a to both sides.

$$-4a+4a=0$$

Combine -4a and 4a to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{C}$$

Variable a cannot be equal to any of the values -1,1,0.

$$a\in \mathrm{C}\setminus -1,0,1$$

Solve for a

$$a\in \mathrm{R}\setminus -1,0,1$$

Solution Steps

Variable a cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by 2a\left(a-1\right)\left(a+1\right), the least common multiple of a^{2}+a,2a-2,2a+2,a\left(a^{2}-1\right).

$$\left(2a-2\right)\left(a-1\right)-\left(a+1\right)a-\left(a-1\right)a=2\left(1-2a\right)$$

Use the distributive property to multiply 2a-2 by a-1 and combine like terms.

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

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

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

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

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

Combine 2a^{2} and -a^{2} to get a^{2}.

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

Combine -4a and -a to get -5a.

$$a^{2}-5a+2-\left(a-1\right)a=2\left(1-2a\right)$$

Use the distributive property to multiply a-1 by a.

$$a^{2}-5a+2-\left(a^{2}-a\right)=2\left(1-2a\right)$$

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

$$a^{2}-5a+2-a^{2}+a=2\left(1-2a\right)$$

Combine a^{2} and -a^{2} to get 0.

$$-5a+2+a=2\left(1-2a\right)$$

Combine -5a and a to get -4a.

$$-4a+2=2\left(1-2a\right)$$

Use the distributive property to multiply 2 by 1-2a.

$$-4a+2=2-4a$$

Subtract 2 from both sides.

$$-4a+2-2=-4a$$

Subtract 2 from 2 to get 0.

$$-4a=-4a$$

Add 4a to both sides.

$$-4a+4a=0$$

Combine -4a and 4a to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{R}$$

Variable a cannot be equal to any of the values -1,1,0.

$$a\in \mathrm{R}\setminus -1,0,1$$