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

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

Solve for y

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

Solution Steps

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

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

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

$$y=\frac{1}{2}x^{2}+x+\frac{1}{2}-6$$

Subtract 6 from \frac{1}{2} to get -\frac{11}{2}.

$$y=\frac{1}{2}x^{2}+x-\frac{11}{2}$$

Solve for x (complex solution)

$$x=-\left(\sqrt{2\left(y+6\right)}+1\right)$$

$$x=\sqrt{2\left(y+6\right)}-1$$

Solve for x

$$x=-\left(\sqrt{2\left(y+6\right)}+1\right)$$

$$x=\sqrt{2\left(y+6\right)}-1\text{, }y\geq -6$$