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

$$y = 1 / 2 ( x + 4 ) ^ { 2 } + 6$$

Solve for y

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

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$y=\frac{1}{2}\left(x^{2}+8x+16\right)+6$$

Use the distributive property to multiply \frac{1}{2} by x^{2}+8x+16.

$$y=\frac{1}{2}x^{2}+4x+8+6$$

Add 8 and 6 to get 14.

$$y=\frac{1}{2}x^{2}+4x+14$$

Solve for x (complex solution)

$$x=-\left(\sqrt{2\left(y-6\right)}+4\right)$$

$$x=\sqrt{2\left(y-6\right)}-4$$

Solve for x

$$x=-\left(\sqrt{2\left(y-6\right)}+4\right)$$

$$x=\sqrt{2\left(y-6\right)}-4\text{, }y\geq 6$$