Solve 1/4a+4-1/8a-8-1/12a^2+12=3a^3-11a^2+3a-7/24(a^4-1)

$$\frac { 1 } { 4 a + 4 } - \frac { 1 } { 8 a - 8 } - \frac { 1 } { 12 a ^ { 2 } + 12 } = \frac { 3 a ^ { 3 } - 11 a ^ { 2 } + 3 a - 7 } { 24 ( a ^ { 4 } - 1 ) }$$

Solve for a (complex solution)

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

Solution Steps

Variable a cannot be equal to any of the values -1,-i,i,1 since division by zero is not defined. Multiply both sides of the equation by 24\left(a-1\right)\left(a+1\right)\left(a-i\right)\left(a+i\right), the least common multiple of 4a+4,8a-8,12a^{2}+12,24\left(a^{4}-1\right).

$$6\left(a-1\right)\left(a+i\right)\left(a-i\right)-3\left(a+1\right)\left(a+i\right)\left(a-i\right)-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Use the distributive property to multiply 6 by a-1.

$$\left(6a-6\right)\left(a+i\right)\left(a-i\right)-3\left(a+1\right)\left(a+i\right)\left(a-i\right)-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Use the distributive property to multiply 6a-6 by a+i and combine like terms.

$$\left(6a^{2}+\left(-6+6i\right)a-6i\right)\left(a-i\right)-3\left(a+1\right)\left(a+i\right)\left(a-i\right)-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Use the distributive property to multiply 6a^{2}+\left(-6+6i\right)a-6i by a-i and combine like terms.

$$6a^{3}-6a^{2}+6a-6-3\left(a+1\right)\left(a+i\right)\left(a-i\right)-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Use the distributive property to multiply -3 by a+1.

$$6a^{3}-6a^{2}+6a-6+\left(-3a-3\right)\left(a+i\right)\left(a-i\right)-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Use the distributive property to multiply -3a-3 by a+i and combine like terms.

$$6a^{3}-6a^{2}+6a-6+\left(-3a^{2}+\left(-3-3i\right)a-3i\right)\left(a-i\right)-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Use the distributive property to multiply -3a^{2}+\left(-3-3i\right)a-3i by a-i and combine like terms.

$$6a^{3}-6a^{2}+6a-6-3a^{3}-3a^{2}-3a-3-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Combine 6a^{3} and -3a^{3} to get 3a^{3}.

$$3a^{3}-6a^{2}+6a-6-3a^{2}-3a-3-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Combine -6a^{2} and -3a^{2} to get -9a^{2}.

$$3a^{3}-9a^{2}+6a-6-3a-3-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Combine 6a and -3a to get 3a.

$$3a^{3}-9a^{2}+3a-6-3-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Subtract 3 from -6 to get -9.

$$3a^{3}-9a^{2}+3a-9-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

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

$$3a^{3}-9a^{2}+3a-9-2a^{2}+2=3a^{3}-11a^{2}+3a-7$$

Combine -9a^{2} and -2a^{2} to get -11a^{2}.

$$3a^{3}-11a^{2}+3a-9+2=3a^{3}-11a^{2}+3a-7$$

Add -9 and 2 to get -7.

$$3a^{3}-11a^{2}+3a-7=3a^{3}-11a^{2}+3a-7$$

Subtract 3a^{3} from both sides.

$$3a^{3}-11a^{2}+3a-7-3a^{3}=-11a^{2}+3a-7$$

Combine 3a^{3} and -3a^{3} to get 0.

$$-11a^{2}+3a-7=-11a^{2}+3a-7$$

Add 11a^{2} to both sides.

$$-11a^{2}+3a-7+11a^{2}=3a-7$$

Combine -11a^{2} and 11a^{2} to get 0.

$$3a-7=3a-7$$

Subtract 3a from both sides.

$$3a-7-3a=-7$$

Combine 3a and -3a to get 0.

$$-7=-7$$

Compare -7 and -7.

$$\text{true}$$

This is true for any a.

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

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

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

Solve for a

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

Solution Steps

Variable a cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by 24\left(a-1\right)\left(a+1\right)\left(a^{2}+1\right), the least common multiple of 4a+4,8a-8,12a^{2}+12,24\left(a^{4}-1\right).

$$6\left(a-1\right)\left(a^{2}+1\right)-3\left(a+1\right)\left(a^{2}+1\right)-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Use the distributive property to multiply 6 by a-1.

$$\left(6a-6\right)\left(a^{2}+1\right)-3\left(a+1\right)\left(a^{2}+1\right)-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Use the distributive property to multiply 6a-6 by a^{2}+1.

$$6a^{3}+6a-6a^{2}-6-3\left(a+1\right)\left(a^{2}+1\right)-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Use the distributive property to multiply -3 by a+1.

$$6a^{3}+6a-6a^{2}-6+\left(-3a-3\right)\left(a^{2}+1\right)-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Use the distributive property to multiply -3a-3 by a^{2}+1.

$$6a^{3}+6a-6a^{2}-6-3a^{3}-3a-3a^{2}-3-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Combine 6a^{3} and -3a^{3} to get 3a^{3}.

$$3a^{3}+6a-6a^{2}-6-3a-3a^{2}-3-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Combine 6a and -3a to get 3a.

$$3a^{3}+3a-6a^{2}-6-3a^{2}-3-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Combine -6a^{2} and -3a^{2} to get -9a^{2}.

$$3a^{3}+3a-9a^{2}-6-3-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

Subtract 3 from -6 to get -9.

$$3a^{3}+3a-9a^{2}-9-\left(2a^{2}-2\right)=3a^{3}-11a^{2}+3a-7$$

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

$$3a^{3}+3a-9a^{2}-9-2a^{2}+2=3a^{3}-11a^{2}+3a-7$$

Combine -9a^{2} and -2a^{2} to get -11a^{2}.

$$3a^{3}+3a-11a^{2}-9+2=3a^{3}-11a^{2}+3a-7$$

Add -9 and 2 to get -7.

$$3a^{3}+3a-11a^{2}-7=3a^{3}-11a^{2}+3a-7$$

Subtract 3a^{3} from both sides.

$$3a^{3}+3a-11a^{2}-7-3a^{3}=-11a^{2}+3a-7$$

Combine 3a^{3} and -3a^{3} to get 0.

$$3a-11a^{2}-7=-11a^{2}+3a-7$$

Add 11a^{2} to both sides.

$$3a-11a^{2}-7+11a^{2}=3a-7$$

Combine -11a^{2} and 11a^{2} to get 0.

$$3a-7=3a-7$$

Subtract 3a from both sides.

$$3a-7-3a=-7$$

Combine 3a and -3a to get 0.

$$-7=-7$$

Compare -7 and -7.

$$\text{true}$$

This is true for any a.

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

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

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