Solve {left(x-1/2right)}^2-2y=1-(2-x)(2+x)

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

Solve for x

$$x=\frac{13}{4}-2y$$

Steps for Solving Linear Equation

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

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

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

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

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

$$x^{2}-x+\frac{1}{4}-2y=1-4+x^{2}$$

Subtract 4 from 1 to get -3.

$$x^{2}-x+\frac{1}{4}-2y=-3+x^{2}$$

Subtract x^{2} from both sides.

$$x^{2}-x+\frac{1}{4}-2y-x^{2}=-3$$

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

$$-x+\frac{1}{4}-2y=-3$$

Subtract \frac{1}{4} from both sides.

$$-x-2y=-3-\frac{1}{4}$$

Subtract \frac{1}{4} from -3 to get -\frac{13}{4}.

$$-x-2y=-\frac{13}{4}$$

Add 2y to both sides.

$$-x=-\frac{13}{4}+2y$$

The equation is in standard form.

$$-x=2y-\frac{13}{4}$$

Divide both sides by -1.

$$\frac{-x}{-1}=\frac{2y-\frac{13}{4}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x=\frac{2y-\frac{13}{4}}{-1}$$

Divide -\frac{13}{4}+2y by -1.

$$x=\frac{13}{4}-2y$$

Solve for y

$$y=-\frac{x}{2}+\frac{13}{8}$$

Steps for Solving Linear Equation

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

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

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

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

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

$$x^{2}-x+\frac{1}{4}-2y=1-4+x^{2}$$

Subtract 4 from 1 to get -3.

$$x^{2}-x+\frac{1}{4}-2y=-3+x^{2}$$

Subtract x^{2} from both sides.

$$-x+\frac{1}{4}-2y=-3+x^{2}-x^{2}$$

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

$$-x+\frac{1}{4}-2y=-3$$

Add x to both sides.

$$\frac{1}{4}-2y=-3+x$$

Subtract \frac{1}{4} from both sides.

$$-2y=-3+x-\frac{1}{4}$$

Subtract \frac{1}{4} from -3 to get -\frac{13}{4}.

$$-2y=-\frac{13}{4}+x$$

The equation is in standard form.

$$-2y=x-\frac{13}{4}$$

Divide both sides by -2.

$$\frac{-2y}{-2}=\frac{x-\frac{13}{4}}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$y=\frac{x-\frac{13}{4}}{-2}$$

Divide -\frac{13}{4}+x by -2.

$$y=-\frac{x}{2}+\frac{13}{8}$$