Solve 1/4(6-x)(x-2)+(x+1/2)^2=x^2-(x-1/2)(x+1/2)

$$\frac { 1 } { 4 } ( 6 - x ) ( x - 2 ) + ( x + \frac { 1 } { 2 } ) ^ { 2 } = x ^ { 2 } - ( \frac { x - 1 } { 2 } ) ( \frac { x + 1 } { 2 } )$$

Solve for x

$$x=1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$1\left(6-x\right)\left(x-2\right)+4\left(x+\frac{1}{2}\right)^{2}=4x^{2}-\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 1 by 6-x.

$$\left(6-x\right)\left(x-2\right)+4\left(x+\frac{1}{2}\right)^{2}=4x^{2}-\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 6-x by x-2 and combine like terms.

$$8x-12-x^{2}+4\left(x+\frac{1}{2}\right)^{2}=4x^{2}-\left(x-1\right)\left(x+1\right)$$

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

$$8x-12-x^{2}+4\left(x^{2}+x+\frac{1}{4}\right)=4x^{2}-\left(x-1\right)\left(x+1\right)$$

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

$$8x-12-x^{2}+4x^{2}+4x+1=4x^{2}-\left(x-1\right)\left(x+1\right)$$

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

$$8x-12+3x^{2}+4x+1=4x^{2}-\left(x-1\right)\left(x+1\right)$$

Combine 8x and 4x to get 12x.

$$12x-12+3x^{2}+1=4x^{2}-\left(x-1\right)\left(x+1\right)$$

Add -12 and 1 to get -11.

$$12x-11+3x^{2}=4x^{2}-\left(x-1\right)\left(x+1\right)$$

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.

$$12x-11+3x^{2}=4x^{2}-\left(x^{2}-1\right)$$

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

$$12x-11+3x^{2}=4x^{2}-x^{2}+1$$

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

$$12x-11+3x^{2}=3x^{2}+1$$

Subtract 3x^{2} from both sides.

$$12x-11+3x^{2}-3x^{2}=1$$

Combine 3x^{2} and -3x^{2} to get 0.

$$12x-11=1$$

Add 11 to both sides.

$$12x=1+11$$

Add 1 and 11 to get 12.

$$12x=12$$

Divide both sides by 12.

$$x=\frac{12}{12}$$

Divide 12 by 12 to get 1.

$$x=1$$