Solve (x/2)^2+(x-4/x)^2=4

$$( \frac { x } { 2 } ) ^ { 2 } + ( x - \frac { 4 } { x } ) ^ { 2 } = 4$$

Solve for x

$$x=2\sqrt{2}\approx 2.828427125$$

$$x=-2\sqrt{2}\approx -2.828427125$$

$$x = -\frac{2 \sqrt{10}}{5} \approx -1.264911064$$

$$x = \frac{2 \sqrt{10}}{5} \approx 1.264911064$$

Steps Using the Quadratic Formula

To raise \frac{x}{2} to a power, raise both numerator and denominator to the power and then divide.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.

$$\frac{x^{2}}{2^{2}}+\left(\frac{xx}{x}-\frac{4}{x}\right)^{2}=4$$

Since \frac{xx}{x} and \frac{4}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}}{2^{2}}+\left(\frac{xx-4}{x}\right)^{2}=4$$

Do the multiplications in xx-4.

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

To raise \frac{x^{2}-4}{x} to a power, raise both numerator and denominator to the power and then divide.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2^{2} and x^{2} is 2^{2}x^{2}. Multiply \frac{x^{2}}{2^{2}} times \frac{x^{2}}{x^{2}}. Multiply \frac{\left(x^{2}-4\right)^{2}}{x^{2}} times \frac{2^{2}}{2^{2}}.

$$\frac{x^{2}x^{2}}{2^{2}x^{2}}+\frac{\left(x^{2}-4\right)^{2}\times 2^{2}}{2^{2}x^{2}}=4$$

Since \frac{x^{2}x^{2}}{2^{2}x^{2}} and \frac{\left(x^{2}-4\right)^{2}\times 2^{2}}{2^{2}x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}x^{2}+\left(x^{2}-4\right)^{2}\times 2^{2}}{2^{2}x^{2}}=4$$

Do the multiplications in x^{2}x^{2}+\left(x^{2}-4\right)^{2}\times 2^{2}.

$$\frac{x^{4}+4x^{4}-32x^{2}+64}{2^{2}x^{2}}=4$$

Combine like terms in x^{4}+4x^{4}-32x^{2}+64.

$$\frac{5x^{4}-32x^{2}+64}{2^{2}x^{2}}=4$$

Calculate 2 to the power of 2 and get 4.

$$\frac{5x^{4}-32x^{2}+64}{4x^{2}}=4$$

Subtract 4 from both sides.

$$\frac{5x^{4}-32x^{2}+64}{4x^{2}}-4=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{4x^{2}}{4x^{2}}.

$$\frac{5x^{4}-32x^{2}+64}{4x^{2}}-\frac{4\times 4x^{2}}{4x^{2}}=0$$

Since \frac{5x^{4}-32x^{2}+64}{4x^{2}} and \frac{4\times 4x^{2}}{4x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5x^{4}-32x^{2}+64-4\times 4x^{2}}{4x^{2}}=0$$

Do the multiplications in 5x^{4}-32x^{2}+64-4\times 4x^{2}.

$$\frac{5x^{4}-32x^{2}+64-16x^{2}}{4x^{2}}=0$$

Combine like terms in 5x^{4}-32x^{2}+64-16x^{2}.

$$\frac{5x^{4}-48x^{2}+64}{4x^{2}}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x^{2}.

$$5x^{4}-48x^{2}+64=0$$

Substitute t for x^{2}.

$$5t^{2}-48t+64=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 5 for a, -48 for b, and 64 for c in the quadratic formula.

$$t=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 5\times 64}}{2\times 5}$$

Do the calculations.

$$t=\frac{48±32}{10}$$

Solve the equation t=\frac{48±32}{10} when ± is plus and when ± is minus.

$$t=8 t=\frac{8}{5}$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=2\sqrt{2} x=-2\sqrt{2} x=\frac{2\sqrt{10}}{5} x=-\frac{2\sqrt{10}}{5}$$