Solve x^2-(x-1)^2=y

$$x ^ { 2 } - ( x - 1 ) ^ { 2 } = y$$

Solve for x

$$x=\frac{y+1}{2}$$

Steps for Solving Linear Equation

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

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

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

$$x^{2}-x^{2}+2x-1=y$$

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

$$2x-1=y$$

Add 1 to both sides.

$$2x=y+1$$

Divide both sides by 2.

$$\frac{2x}{2}=\frac{y+1}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x=\frac{y+1}{2}$$

Solve for y

$$y=2x-1$$

Solution Steps

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

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

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

$$x^{2}-x^{2}+2x-1=y$$

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

$$2x-1=y$$

Swap sides so that all variable terms are on the left hand side.

$$y=2x-1$$