Solve (4)(2x-1)^2+(x-2)(x+2)-4x(x-1/2)

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

Evaluate

$$x\left(13x-14\right)$$

Solution Steps

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

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

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

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

Consider \left(x-2\right)\left(x+2\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 2.

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

Combine 16x^{2} and x^{2} to get 17x^{2}.

$$17x^{2}-16x+4-4-4x\left(x-\frac{1}{2}\right)$$

Subtract 4 from 4 to get 0.

$$17x^{2}-16x-4x\left(x-\frac{1}{2}\right)$$

Use the distributive property to multiply -4x by x-\frac{1}{2}.

$$17x^{2}-16x-4x^{2}+2x$$

Combine 17x^{2} and -4x^{2} to get 13x^{2}.

$$13x^{2}-16x+2x$$

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

$$13x^{2}-14x$$

Expand

$$13x^{2}-14x$$

Solution Steps

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

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

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

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

Consider \left(x-2\right)\left(x+2\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 2.

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

Combine 16x^{2} and x^{2} to get 17x^{2}.

$$17x^{2}-16x+4-4-4x\left(x-\frac{1}{2}\right)$$

Subtract 4 from 4 to get 0.

$$17x^{2}-16x-4x\left(x-\frac{1}{2}\right)$$

Use the distributive property to multiply -4x by x-\frac{1}{2}.

$$17x^{2}-16x-4x^{2}+2x$$

Combine 17x^{2} and -4x^{2} to get 13x^{2}.

$$13x^{2}-16x+2x$$

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

$$13x^{2}-14x$$