Solve 2k=sqrt{k^2+1}

$$2 k = \sqrt { k ^ { 2 } + 1 }$$

Solve for k

$$k=\frac{\sqrt{3}}{3}\approx 0.577350269$$

Solution Steps

Square both sides of the equation.

$$\left(2k\right)^{2}=\left(\sqrt{k^{2}+1}\right)^{2}$$

Expand \left(2k\right)^{2}.

$$2^{2}k^{2}=\left(\sqrt{k^{2}+1}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4k^{2}=\left(\sqrt{k^{2}+1}\right)^{2}$$

Calculate \sqrt{k^{2}+1} to the power of 2 and get k^{2}+1.

$$4k^{2}=k^{2}+1$$

Subtract k^{2} from both sides.

$$4k^{2}-k^{2}=1$$

Combine 4k^{2} and -k^{2} to get 3k^{2}.

$$3k^{2}=1$$

Divide both sides by 3.

$$k^{2}=\frac{1}{3}$$

Take the square root of both sides of the equation.

$$k=\frac{\sqrt{3}}{3} k=-\frac{\sqrt{3}}{3}$$

Substitute \frac{\sqrt{3}}{3} for k in the equation 2k=\sqrt{k^{2}+1}.

$$2\times \frac{\sqrt{3}}{3}=\sqrt{\left(\frac{\sqrt{3}}{3}\right)^{2}+1}$$

Simplify. The value k=\frac{\sqrt{3}}{3} satisfies the equation.

$$\frac{2}{3}\times 3^{\frac{1}{2}}=\frac{2}{3}\times 3^{\frac{1}{2}}$$

Substitute -\frac{\sqrt{3}}{3} for k in the equation 2k=\sqrt{k^{2}+1}.

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

Simplify. The value k=-\frac{\sqrt{3}}{3} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{2}{3}\times 3^{\frac{1}{2}}=\frac{2}{3}\times 3^{\frac{1}{2}}$$

Equation 2k=\sqrt{k^{2}+1} has a unique solution.

$$k=\frac{\sqrt{3}}{3}$$