Solve x^2-2sqrt{3}kx+k+1=x^2-2x(sqrt{3}k)+(k+1)

$$x ^ { 2 } - 2 \sqrt { 3 } k x + k + 1 = x ^ { 2 } - 2 x ( \sqrt { 3 } k ) + ( k + 1 )$$

Solve for k (complex solution)

$$k\in \mathrm{C}$$

Solution Steps

Subtract x^{2}-2x\sqrt{3}k from both sides.

$$x^{2}-2\sqrt{3}kx+k+1-\left(x^{2}-2x\sqrt{3}k\right)=k+1$$

Combine x^{2}-2\sqrt{3}kx and -\left(x^{2}-2x\sqrt{3}k\right) to get 0.

$$k+1=k+1$$

Subtract k from both sides.

$$k+1-k=1$$

Combine k and -k to get 0.

$$1=1$$

Compare 1 and 1.

$$\text{true}$$

This is true for any k.

$$k\in \mathrm{C}$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

Subtract x^{2}-2x\sqrt{3}k from both sides.

$$x^{2}-2\sqrt{3}kx+k+1-\left(x^{2}-2x\sqrt{3}k\right)=k+1$$

Combine x^{2}-2\sqrt{3}kx and -\left(x^{2}-2x\sqrt{3}k\right) to get 0.

$$k+1=k+1$$

Reorder the terms.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for k

$$k\in \mathrm{R}$$

Solution Steps

Subtract x^{2}-2x\sqrt{3}k from both sides.

$$x^{2}-2\sqrt{3}kx+k+1-\left(x^{2}-2x\sqrt{3}k\right)=k+1$$

Combine x^{2}-2\sqrt{3}kx and -\left(x^{2}-2x\sqrt{3}k\right) to get 0.

$$k+1=k+1$$

Subtract k from both sides.

$$k+1-k=1$$

Combine k and -k to get 0.

$$1=1$$

Compare 1 and 1.

$$\text{true}$$

This is true for any k.

$$k\in \mathrm{R}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

Subtract x^{2}-2x\sqrt{3}k from both sides.

$$x^{2}-2\sqrt{3}kx+k+1-\left(x^{2}-2x\sqrt{3}k\right)=k+1$$

Combine x^{2}-2\sqrt{3}kx and -\left(x^{2}-2x\sqrt{3}k\right) to get 0.

$$k+1=k+1$$

Reorder the terms.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$