Solve f(x)=x-(x-1)^2-(x+1)(1-x)

$$f ( x ) = x - ( x - 1 ) ^ { 2 } - ( x + 1 ) ( 1 - x )$$

Evaluate

$$3x-2$$

Solution Steps

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

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

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

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

Combine x and 2x to get 3x.

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

Consider \left(x+1\right)\left(1-x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

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

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

Subtract 1 from -1 to get -2.

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

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

$$3x-2$$

Expand

$$3x-2$$

Solution Steps

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

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

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

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

Combine x and 2x to get 3x.

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

Consider \left(x+1\right)\left(1-x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

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

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

Subtract 1 from -1 to get -2.

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

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

$$3x-2$$