Solve x^2+(16/x)^2=64

$$x ^ { 2 } + ( \frac { 16 } { x } ) ^ { 2 } = 64$$

Solve for x

$$x=2\sqrt{6}-2\sqrt{2}\approx 2.070552361$$

$$x=2\sqrt{2}-2\sqrt{6}\approx -2.070552361$$

$$x=-2\sqrt{2}-2\sqrt{6}\approx -7.72740661$$

$$x=2\sqrt{2}+2\sqrt{6}\approx 7.72740661$$

Steps Using the Quadratic Formula

To raise \frac{16}{x} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}+\frac{16^{2}}{x^{2}}=64$$

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

$$\frac{x^{2}x^{2}}{x^{2}}+\frac{16^{2}}{x^{2}}=64$$

Since \frac{x^{2}x^{2}}{x^{2}} and \frac{16^{2}}{x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}x^{2}+16^{2}}{x^{2}}=64$$

Do the multiplications in x^{2}x^{2}+16^{2}.

$$\frac{x^{4}+16^{2}}{x^{2}}=64$$

Combine like terms in x^{4}+16^{2}.

$$\frac{x^{4}+256}{x^{2}}=64$$

Subtract 64 from both sides.

$$\frac{x^{4}+256}{x^{2}}-64=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 64 times \frac{x^{2}}{x^{2}}.

$$\frac{x^{4}+256}{x^{2}}-\frac{64x^{2}}{x^{2}}=0$$

Since \frac{x^{4}+256}{x^{2}} and \frac{64x^{2}}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{4}+256-64x^{2}}{x^{2}}=0$$

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

$$x^{4}+256-64x^{2}=0$$

Substitute t for x^{2}.

$$t^{2}-64t+256=0$$

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 1 for a, -64 for b, and 256 for c in the quadratic formula.

$$t=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 1\times 256}}{2}$$

Do the calculations.

$$t=\frac{64±32\sqrt{3}}{2}$$

Solve the equation t=\frac{64±32\sqrt{3}}{2} when ± is plus and when ± is minus.

$$t=16\sqrt{3}+32 t=32-16\sqrt{3}$$

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

$$x=2\sqrt{2}+2\sqrt{6} x=-\left(2\sqrt{2}+2\sqrt{6}\right) x=-\left(2\sqrt{2}-2\sqrt{6}\right) x=2\sqrt{2}-2\sqrt{6}$$