Solve (x+1)^2+y=1/4

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

Solve for y

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

Solution Steps

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

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

Subtract x^{2} from both sides.

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

Subtract 2x from both sides.

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

Subtract 1 from both sides.

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

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

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

Solve for x (complex solution)

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

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

Solve for x

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

$$x=\frac{\sqrt{1-4y}}{2}-1\text{, }y\leq \frac{1}{4}$$