Solve (2x-1)^2-(x-2)^2+3(1-x)^2

$$( 2 x - 1 ) ^ { 2 } - ( x - 2 ) ^ { 2 } + 3 ( 1 - x ) ^ { 2 }$$

Evaluate

$$6x\left(x-1\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-1\right)^{2}.

$$4x^{2}-4x+1-\left(x-2\right)^{2}+3\left(1-x\right)^{2}$$

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

$$4x^{2}-4x+1-\left(x^{2}-4x+4\right)+3\left(1-x\right)^{2}$$

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

$$4x^{2}-4x+1-x^{2}+4x-4+3\left(1-x\right)^{2}$$

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

$$3x^{2}-4x+1+4x-4+3\left(1-x\right)^{2}$$

Combine -4x and 4x to get 0.

$$3x^{2}+1-4+3\left(1-x\right)^{2}$$

Subtract 4 from 1 to get -3.

$$3x^{2}-3+3\left(1-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-x\right)^{2}.

$$3x^{2}-3+3\left(1-2x+x^{2}\right)$$

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

$$3x^{2}-3+3-6x+3x^{2}$$

Add -3 and 3 to get 0.

$$3x^{2}-6x+3x^{2}$$

Combine 3x^{2} and 3x^{2} to get 6x^{2}.

$$6x^{2}-6x$$

Expand

$$6x^{2}-6x$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-1\right)^{2}.

$$4x^{2}-4x+1-\left(x-2\right)^{2}+3\left(1-x\right)^{2}$$

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

$$4x^{2}-4x+1-\left(x^{2}-4x+4\right)+3\left(1-x\right)^{2}$$

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

$$4x^{2}-4x+1-x^{2}+4x-4+3\left(1-x\right)^{2}$$

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

$$3x^{2}-4x+1+4x-4+3\left(1-x\right)^{2}$$

Combine -4x and 4x to get 0.

$$3x^{2}+1-4+3\left(1-x\right)^{2}$$

Subtract 4 from 1 to get -3.

$$3x^{2}-3+3\left(1-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-x\right)^{2}.

$$3x^{2}-3+3\left(1-2x+x^{2}\right)$$

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

$$3x^{2}-3+3-6x+3x^{2}$$

Add -3 and 3 to get 0.

$$3x^{2}-6x+3x^{2}$$

Combine 3x^{2} and 3x^{2} to get 6x^{2}.

$$6x^{2}-6x$$