Solve (a+b)*(a-b)=a^2-b^2

$$( a + b ) \cdot ( a - b ) = a ^ { 2 } - b ^ { 2 }$$

Solve for a (complex solution)

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

Solution Steps

Consider \left(a+b\right)\left(a-b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Subtract a^{2} from both sides.

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

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

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

Cancel out -1 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

Consider \left(a+b\right)\left(a-b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Add b^{2} to both sides.

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

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

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

Reorder the terms.

$$\text{true}$$

This is true for any b.

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

Solve for a

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

Solution Steps

Consider \left(a+b\right)\left(a-b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Subtract a^{2} from both sides.

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

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

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

Cancel out -1 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

Consider \left(a+b\right)\left(a-b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Add b^{2} to both sides.

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

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

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

Reorder the terms.

$$\text{true}$$

This is true for any b.

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