Solve y=4(x-1)^2+k

$$y = 4 ( x - 1 ) ^ { 2 } + k$$

Solve for k

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

Solution Steps

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

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

Use the distributive property to multiply 4 by x^{2}-2x+1.

$$y=4x^{2}-8x+4+k$$

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

$$4x^{2}-8x+4+k=y$$

Subtract 4x^{2} from both sides.

$$-8x+4+k=y-4x^{2}$$

Add 8x to both sides.

$$4+k=y-4x^{2}+8x$$

Subtract 4 from both sides.

$$k=y-4x^{2}+8x-4$$

Solve for x (complex solution)

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

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

Solve for x

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

$$x=\frac{\sqrt{y-k}}{2}+1\text{, }y\geq k$$