Solve (a^2-b^2)+3(a+b)(a-b)=4(a^2-b^2)

$$( a ^ { 2 } - b ^ { 2 } ) + 3 ( a + b ) ( a - b ) = 4 ( a ^ { 2 } - b ^ { 2 } )$$

Solve for a (complex solution)

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

Solution Steps

Use the distributive property to multiply 3 by a+b.

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

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

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

Combine a^{2} and 3a^{2} to get 4a^{2}.

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

Combine -b^{2} and -3b^{2} to get -4b^{2}.

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

Use the distributive property to multiply 4 by a^{2}-b^{2}.

$$4a^{2}-4b^{2}=4a^{2}-4b^{2}$$

Subtract 4a^{2} from both sides.

$$4a^{2}-4b^{2}-4a^{2}=-4b^{2}$$

Combine 4a^{2} and -4a^{2} to get 0.

$$-4b^{2}=-4b^{2}$$

Cancel out -4 on both sides.

$$b^{2}=b^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any a.

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

Solve for b (complex solution)

$$b\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply 3 by a+b.

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

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

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

Combine a^{2} and 3a^{2} to get 4a^{2}.

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

Combine -b^{2} and -3b^{2} to get -4b^{2}.

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

Use the distributive property to multiply 4 by a^{2}-b^{2}.

$$4a^{2}-4b^{2}=4a^{2}-4b^{2}$$

Add 4b^{2} to both sides.

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

Combine -4b^{2} and 4b^{2} to get 0.

$$4a^{2}=4a^{2}$$

Cancel out 4 on both sides.

$$a^{2}=a^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any b.

$$b\in \mathrm{C}$$

Solve for a

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

Solution Steps

Use the distributive property to multiply 3 by a+b.

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

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

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

Combine a^{2} and 3a^{2} to get 4a^{2}.

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

Combine -b^{2} and -3b^{2} to get -4b^{2}.

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

Use the distributive property to multiply 4 by a^{2}-b^{2}.

$$4a^{2}-4b^{2}=4a^{2}-4b^{2}$$

Subtract 4a^{2} from both sides.

$$4a^{2}-4b^{2}-4a^{2}=-4b^{2}$$

Combine 4a^{2} and -4a^{2} to get 0.

$$-4b^{2}=-4b^{2}$$

Cancel out -4 on both sides.

$$b^{2}=b^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any a.

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

Solve for b

$$b\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply 3 by a+b.

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

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

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

Combine a^{2} and 3a^{2} to get 4a^{2}.

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

Combine -b^{2} and -3b^{2} to get -4b^{2}.

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

Use the distributive property to multiply 4 by a^{2}-b^{2}.

$$4a^{2}-4b^{2}=4a^{2}-4b^{2}$$

Add 4b^{2} to both sides.

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

Combine -4b^{2} and 4b^{2} to get 0.

$$4a^{2}=4a^{2}$$

Cancel out 4 on both sides.

$$a^{2}=a^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any b.

$$b\in \mathrm{R}$$