Solve x^2+(3/x)^2=25

$$x ^ { 2 } + ( \frac { 3 } { x } ) ^ { 2 } = 25$$

Solve for x

$$x=\frac{\sqrt{31}-\sqrt{19}}{2}\approx 0.60443271$$

$$x=\frac{\sqrt{19}-\sqrt{31}}{2}\approx -0.60443271$$

$$x=\frac{-\sqrt{19}-\sqrt{31}}{2}\approx -4.963331653$$

$$x = \frac{\sqrt{19} + \sqrt{31}}{2} \approx 4.963331653$$

Steps Using the Quadratic Formula

To raise \frac{3}{x} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}+\frac{3^{2}}{x^{2}}=25$$

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

$$\frac{x^{2}x^{2}}{x^{2}}+\frac{3^{2}}{x^{2}}=25$$

Since \frac{x^{2}x^{2}}{x^{2}} and \frac{3^{2}}{x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}x^{2}+3^{2}}{x^{2}}=25$$

Do the multiplications in x^{2}x^{2}+3^{2}.

$$\frac{x^{4}+3^{2}}{x^{2}}=25$$

Combine like terms in x^{4}+3^{2}.

$$\frac{x^{4}+9}{x^{2}}=25$$

Subtract 25 from both sides.

$$\frac{x^{4}+9}{x^{2}}-25=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 25 times \frac{x^{2}}{x^{2}}.

$$\frac{x^{4}+9}{x^{2}}-\frac{25x^{2}}{x^{2}}=0$$

Since \frac{x^{4}+9}{x^{2}} and \frac{25x^{2}}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{4}+9-25x^{2}}{x^{2}}=0$$

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

$$x^{4}+9-25x^{2}=0$$

Substitute t for x^{2}.

$$t^{2}-25t+9=0$$

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, -25 for b, and 9 for c in the quadratic formula.

$$t=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 1\times 9}}{2}$$

Do the calculations.

$$t=\frac{25±\sqrt{589}}{2}$$

Solve the equation t=\frac{25±\sqrt{589}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{589}+25}{2} t=\frac{25-\sqrt{589}}{2}$$

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

$$x=\frac{\sqrt{19}+\sqrt{31}}{2} x=-\frac{\sqrt{19}+\sqrt{31}}{2} x=-\frac{\sqrt{19}-\sqrt{31}}{2} x=\frac{\sqrt{19}-\sqrt{31}}{2}$$