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

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

Evaluate

$$x\left(x-2\right)$$

Solution Steps

Consider \left(x+1\right)\left(x-1\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.

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

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

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

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$5x^{2}-1-4x+1-2x\left(2x-1\right)$$

Add -1 and 1 to get 0.

$$5x^{2}-4x-2x\left(2x-1\right)$$

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

$$5x^{2}-4x-4x^{2}+2x$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

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

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

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

Expand

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

Solution Steps

Consider \left(x+1\right)\left(x-1\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.

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

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

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

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$5x^{2}-1-4x+1-2x\left(2x-1\right)$$

Add -1 and 1 to get 0.

$$5x^{2}-4x-2x\left(2x-1\right)$$

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

$$5x^{2}-4x-4x^{2}+2x$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

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

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

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