Solve 1/4=(2sqrt{1-k^2/4})^2

$$\frac { 1 } { 4 } = ( 2 \sqrt { 1 - \frac { k ^ { 2 } } { 4 } } ) ^ { 2 }$$

Solve for k

$$k = \frac{\sqrt{15}}{2} \approx 1.936491673$$

$$k = -\frac{\sqrt{15}}{2} \approx -1.936491673$$

Steps by Finding Square Root

Expand \left(2\sqrt{1-\frac{k^{2}}{4}}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

Calculate \sqrt{1-\frac{k^{2}}{4}} to the power of 2 and get 1-\frac{k^{2}}{4}.

$$\frac{1}{4}=4\left(1-\frac{k^{2}}{4}\right)$$

Use the distributive property to multiply 4 by 1-\frac{k^{2}}{4}.

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

Express 4\left(-\frac{k^{2}}{4}\right) as a single fraction.

$$\frac{1}{4}=4+\frac{-4k^{2}}{4}$$

Cancel out 4 and 4.

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

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

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

Subtract 4 from both sides.

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

Subtract 4 from \frac{1}{4} to get -\frac{15}{4}.

$$-k^{2}=-\frac{15}{4}$$

Divide both sides by -1.

$$k^{2}=\frac{-\frac{15}{4}}{-1}$$

Express \frac{-\frac{15}{4}}{-1} as a single fraction.

$$k^{2}=\frac{-15}{4\left(-1\right)}$$

Multiply 4 and -1 to get -4.

$$k^{2}=\frac{-15}{-4}$$

Fraction \frac{-15}{-4} can be simplified to \frac{15}{4} by removing the negative sign from both the numerator and the denominator.

$$k^{2}=\frac{15}{4}$$

Take the square root of both sides of the equation.

$$k=\frac{\sqrt{15}}{2} k=-\frac{\sqrt{15}}{2}$$

Steps Using the Quadratic Formula

Expand \left(2\sqrt{1-\frac{k^{2}}{4}}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

Calculate \sqrt{1-\frac{k^{2}}{4}} to the power of 2 and get 1-\frac{k^{2}}{4}.

$$\frac{1}{4}=4\left(1-\frac{k^{2}}{4}\right)$$

Use the distributive property to multiply 4 by 1-\frac{k^{2}}{4}.

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

Express 4\left(-\frac{k^{2}}{4}\right) as a single fraction.

$$\frac{1}{4}=4+\frac{-4k^{2}}{4}$$

Cancel out 4 and 4.

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

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

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

Subtract \frac{1}{4} from both sides.

$$4-k^{2}-\frac{1}{4}=0$$

Subtract \frac{1}{4} from 4 to get \frac{15}{4}.

$$\frac{15}{4}-k^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-k^{2}+\frac{15}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0 for b, and \frac{15}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{0±\sqrt{0^{2}-4\left(-1\right)\times \frac{15}{4}}}{2\left(-1\right)}$$

Square 0.

$$k=\frac{0±\sqrt{-4\left(-1\right)\times \frac{15}{4}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$k=\frac{0±\sqrt{4\times \frac{15}{4}}}{2\left(-1\right)}$$

Multiply 4 times \frac{15}{4}.

$$k=\frac{0±\sqrt{15}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$k=\frac{0±\sqrt{15}}{-2}$$

Now solve the equation k=\frac{0±\sqrt{15}}{-2} when ± is plus.

$$k=-\frac{\sqrt{15}}{2}$$

Now solve the equation k=\frac{0±\sqrt{15}}{-2} when ± is minus.

$$k=\frac{\sqrt{15}}{2}$$

The equation is now solved.

$$k=-\frac{\sqrt{15}}{2} k=\frac{\sqrt{15}}{2}$$