Solve (5/y)^2-y^2=16

$$( \frac { 5 } { y } ) ^ { 2 } - y ^ { 2 } = 16$$

Solve for y (complex solution)

$$y=-i\sqrt{\sqrt{89}+8}\approx -0-4.175401913i$$

$$y=i\sqrt{\sqrt{89}+8}\approx 4.175401913i$$

$$y=\sqrt{\sqrt{89}-8}\approx 1.197489512$$

$$y=-\sqrt{\sqrt{89}-8}\approx -1.197489512$$

Steps Using the Quadratic Formula

To raise \frac{5}{y} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{5^{2}}{y^{2}}-y^{2}=16$$

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

$$\frac{5^{2}}{y^{2}}-\frac{y^{2}y^{2}}{y^{2}}=16$$

Since \frac{5^{2}}{y^{2}} and \frac{y^{2}y^{2}}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5^{2}-y^{2}y^{2}}{y^{2}}=16$$

Do the multiplications in 5^{2}-y^{2}y^{2}.

$$\frac{5^{2}-y^{4}}{y^{2}}=16$$

Combine like terms in 5^{2}-y^{4}.

$$\frac{25-y^{4}}{y^{2}}=16$$

Subtract 16 from both sides.

$$\frac{25-y^{4}}{y^{2}}-16=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 16 times \frac{y^{2}}{y^{2}}.

$$\frac{25-y^{4}}{y^{2}}-\frac{16y^{2}}{y^{2}}=0$$

Since \frac{25-y^{4}}{y^{2}} and \frac{16y^{2}}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{25-y^{4}-16y^{2}}{y^{2}}=0$$

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

$$25-y^{4}-16y^{2}=0$$

Substitute t for y^{2}.

$$-t^{2}-16t+25=0$$

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, -16 for b, and 25 for c in the quadratic formula.

$$t=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-1\right)\times 25}}{-2}$$

Do the calculations.

$$t=\frac{16±2\sqrt{89}}{-2}$$

Solve the equation t=\frac{16±2\sqrt{89}}{-2} when ± is plus and when ± is minus.

$$t=-\sqrt{89}-8 t=\sqrt{89}-8$$

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

$$y=-i\sqrt{\sqrt{89}+8} y=i\sqrt{\sqrt{89}+8} y=-\sqrt{\sqrt{89}-8} y=\sqrt{\sqrt{89}-8}$$

Solve for y

$$y=\sqrt{\sqrt{89}-8}\approx 1.197489512$$

$$y=-\sqrt{\sqrt{89}-8}\approx -1.197489512$$

Steps Using the Quadratic Formula

To raise \frac{5}{y} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{5^{2}}{y^{2}}-y^{2}=16$$

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

$$\frac{5^{2}}{y^{2}}-\frac{y^{2}y^{2}}{y^{2}}=16$$

Since \frac{5^{2}}{y^{2}} and \frac{y^{2}y^{2}}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5^{2}-y^{2}y^{2}}{y^{2}}=16$$

Do the multiplications in 5^{2}-y^{2}y^{2}.

$$\frac{5^{2}-y^{4}}{y^{2}}=16$$

Combine like terms in 5^{2}-y^{4}.

$$\frac{25-y^{4}}{y^{2}}=16$$

Subtract 16 from both sides.

$$\frac{25-y^{4}}{y^{2}}-16=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 16 times \frac{y^{2}}{y^{2}}.

$$\frac{25-y^{4}}{y^{2}}-\frac{16y^{2}}{y^{2}}=0$$

Since \frac{25-y^{4}}{y^{2}} and \frac{16y^{2}}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{25-y^{4}-16y^{2}}{y^{2}}=0$$

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

$$25-y^{4}-16y^{2}=0$$

Substitute t for y^{2}.

$$-t^{2}-16t+25=0$$

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, -16 for b, and 25 for c in the quadratic formula.

$$t=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-1\right)\times 25}}{-2}$$

Do the calculations.

$$t=\frac{16±2\sqrt{89}}{-2}$$

Solve the equation t=\frac{16±2\sqrt{89}}{-2} when ± is plus and when ± is minus.

$$t=-\sqrt{89}-8 t=\sqrt{89}-8$$

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

$$y=\sqrt{\sqrt{89}-8} y=-\sqrt{\sqrt{89}-8}$$