Solve left(4-2xright)^2-left(x-1right)^2+2left(2a-1right)^2

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

Evaluate

$$3x^{2}-14x+8a^{2}-8a+17$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-2x\right)^{2}.

$$16-16x+4x^{2}-\left(x-1\right)^{2}+2\left(2a-1\right)^{2}$$

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

$$16-16x+4x^{2}-\left(x^{2}-2x+1\right)+2\left(2a-1\right)^{2}$$

To find the opposite of x^{2}-2x+1, find the opposite of each term.

$$16-16x+4x^{2}-x^{2}+2x-1+2\left(2a-1\right)^{2}$$

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

$$16-16x+3x^{2}+2x-1+2\left(2a-1\right)^{2}$$

Combine -16x and 2x to get -14x.

$$16-14x+3x^{2}-1+2\left(2a-1\right)^{2}$$

Subtract 1 from 16 to get 15.

$$15-14x+3x^{2}+2\left(2a-1\right)^{2}$$

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

$$15-14x+3x^{2}+2\left(4a^{2}-4a+1\right)$$

Use the distributive property to multiply 2 by 4a^{2}-4a+1.

$$15-14x+3x^{2}+8a^{2}-8a+2$$

Add 15 and 2 to get 17.

$$17-14x+3x^{2}+8a^{2}-8a$$

Expand

$$3x^{2}-14x+8a^{2}-8a+17$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-2x\right)^{2}.

$$16-16x+4x^{2}-\left(x-1\right)^{2}+2\left(2a-1\right)^{2}$$

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

$$16-16x+4x^{2}-\left(x^{2}-2x+1\right)+2\left(2a-1\right)^{2}$$

To find the opposite of x^{2}-2x+1, find the opposite of each term.

$$16-16x+4x^{2}-x^{2}+2x-1+2\left(2a-1\right)^{2}$$

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

$$16-16x+3x^{2}+2x-1+2\left(2a-1\right)^{2}$$

Combine -16x and 2x to get -14x.

$$16-14x+3x^{2}-1+2\left(2a-1\right)^{2}$$

Subtract 1 from 16 to get 15.

$$15-14x+3x^{2}+2\left(2a-1\right)^{2}$$

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

$$15-14x+3x^{2}+2\left(4a^{2}-4a+1\right)$$

Use the distributive property to multiply 2 by 4a^{2}-4a+1.

$$15-14x+3x^{2}+8a^{2}-8a+2$$

Add 15 and 2 to get 17.

$$17-14x+3x^{2}+8a^{2}-8a$$