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

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

Evaluate

$$-x^{2}+7x-8$$

Solution Steps

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

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

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

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

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

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

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

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

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

$$2x-1-x^{2}+4x-4+x-3$$

Combine 2x and 4x to get 6x.

$$6x-1-x^{2}-4+x-3$$

Subtract 4 from -1 to get -5.

$$6x-5-x^{2}+x-3$$

Combine 6x and x to get 7x.

$$7x-5-x^{2}-3$$

Subtract 3 from -5 to get -8.

$$7x-8-x^{2}$$

Expand

$$-x^{2}+7x-8$$

Solution Steps

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

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

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

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

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

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

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

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

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

$$2x-1-x^{2}+4x-4+x-3$$

Combine 2x and 4x to get 6x.

$$6x-1-x^{2}-4+x-3$$

Subtract 4 from -1 to get -5.

$$6x-5-x^{2}+x-3$$

Combine 6x and x to get 7x.

$$7x-5-x^{2}-3$$

Subtract 3 from -5 to get -8.

$$7x-8-x^{2}$$