Solve (a-1)^2-(a-2)^2+2a*(1-a)

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

Evaluate

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

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-1\right)^{2}.

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

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

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

To find the opposite of a^{2}-4a+4, find the opposite of each term.

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

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

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

Combine -2a and 4a to get 2a.

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

Subtract 4 from 1 to get -3.

$$2a-3+2a\left(1-a\right)$$

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

$$2a-3+2a-2a^{2}$$

Combine 2a and 2a to get 4a.

$$4a-3-2a^{2}$$

Expand

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

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-1\right)^{2}.

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

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

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

To find the opposite of a^{2}-4a+4, find the opposite of each term.

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

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

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

Combine -2a and 4a to get 2a.

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

Subtract 4 from 1 to get -3.

$$2a-3+2a\left(1-a\right)$$

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

$$2a-3+2a-2a^{2}$$

Combine 2a and 2a to get 4a.

$$4a-3-2a^{2}$$