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

$$2 ( a - b ) ^ { 2 } - ( a + b ) ( a - b )$$

Evaluate

$$\left(a-b\right)\left(a-3b\right)$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-b\right)^{2}.

$$2\left(a^{2}-2ab+b^{2}\right)-\left(a+b\right)\left(a-b\right)$$

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

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

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}.

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

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

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

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

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

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

$$a^{2}-4ab+3b^{2}$$

Expand

$$a^{2}-4ab+3b^{2}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-b\right)^{2}.

$$2\left(a^{2}-2ab+b^{2}\right)-\left(a+b\right)\left(a-b\right)$$

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

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

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}.

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

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

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

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

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

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

$$a^{2}-4ab+3b^{2}$$