Solve (12/y)^2+5y^2=16

$$( \frac { 12 } { y } ) ^ { 2 } + 5 y ^ { 2 } = 16$$

Solve for y

$$y=\frac{2\sqrt{3}\times 5^{\frac{3}{4}}e^{-\frac{\arctan(\frac{\sqrt{41}}{2})i}{2}}}{5}\approx 1.866355157-1.372327065i$$

$$y=\frac{2\sqrt{3}\times 5^{\frac{3}{4}}e^{\frac{-\arctan(\frac{\sqrt{41}}{2})i+2\pi i}{2}}}{5}\approx -1.866355157+1.372327065i$$

$$y=\frac{2\sqrt{3}\times 5^{\frac{3}{4}}e^{\frac{\arctan(\frac{\sqrt{41}}{2})i}{2}}}{5}\approx 1.866355157+1.372327065i$$

$$y=\frac{2\sqrt{3}\times 5^{\frac{3}{4}}e^{\frac{\arctan(\frac{\sqrt{41}}{2})i+2\pi i}{2}}}{5}\approx -1.866355157-1.372327065i$$

Steps Using the Quadratic Formula

To raise \frac{12}{y} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{12^{2}}{y^{2}}+5y^{2}=16$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5y^{2} times \frac{y^{2}}{y^{2}}.

$$\frac{12^{2}}{y^{2}}+\frac{5y^{2}y^{2}}{y^{2}}=16$$

Since \frac{12^{2}}{y^{2}} and \frac{5y^{2}y^{2}}{y^{2}} have the same denominator, add them by adding their numerators.

$$\frac{12^{2}+5y^{2}y^{2}}{y^{2}}=16$$

Do the multiplications in 12^{2}+5y^{2}y^{2}.

$$\frac{12^{2}+5y^{4}}{y^{2}}=16$$

Combine like terms in 12^{2}+5y^{4}.

$$\frac{144+5y^{4}}{y^{2}}=16$$

Subtract 16 from both sides.

$$\frac{144+5y^{4}}{y^{2}}-16=0$$

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

$$\frac{144+5y^{4}}{y^{2}}-\frac{16y^{2}}{y^{2}}=0$$

Since \frac{144+5y^{4}}{y^{2}} and \frac{16y^{2}}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{144+5y^{4}-16y^{2}}{y^{2}}=0$$

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

$$144+5y^{4}-16y^{2}=0$$

Substitute t for y^{2}.

$$5t^{2}-16t+144=0$$

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, -16 for b, and 144 for c in the quadratic formula.

$$t=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 5\times 144}}{2\times 5}$$

Do the calculations.

$$t=\frac{16±\sqrt{-2624}}{10}$$

Solve the equation t=\frac{16±\sqrt{-2624}}{10} when ± is plus and when ± is minus.

$$t=\frac{8+4\sqrt{41}i}{5} t=\frac{-4\sqrt{41}i+8}{5}$$

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

$$y=\frac{2\sqrt{3}\times 5^{\frac{3}{4}}e^{\frac{\arctan(\frac{\sqrt{41}}{2})i+2\pi i}{2}}}{5} y=\frac{2\sqrt{3}\times 5^{\frac{3}{4}}e^{\frac{\arctan(\frac{\sqrt{41}}{2})i}{2}}}{5} y=\frac{2\sqrt{3}\times 5^{\frac{3}{4}}e^{-\frac{\arctan(\frac{\sqrt{41}}{2})i}{2}}}{5} y=\frac{2\sqrt{3}\times 5^{\frac{3}{4}}e^{\frac{-\arctan(\frac{\sqrt{41}}{2})i+2\pi i}{2}}}{5}$$

Variable y cannot be equal to 0.

$$y=\frac{2\sqrt{3}\times 5^{\frac{3}{4}}e^{\frac{-\arctan(\frac{\sqrt{41}}{2})i+2\pi i}{2}}}{5}\text{, }y\neq 0 y=\frac{2\sqrt{3}\times 5^{\frac{3}{4}}e^{-\frac{\arctan(\frac{\sqrt{41}}{2})i}{2}}}{5}\text{, }y\neq 0 y=\frac{2\sqrt{3}\times 5^{\frac{3}{4}}e^{\frac{\arctan(\frac{\sqrt{41}}{2})i}{2}}}{5}\text{, }y\neq 0 y=\frac{2\sqrt{3}\times 5^{\frac{3}{4}}e^{\frac{\arctan(\frac{\sqrt{41}}{2})i+2\pi i}{2}}}{5}\text{, }y\neq 0$$