Solve x^2+(15/x)^2=100

$$x ^ { 2 } + ( \frac { 15 } { x } ) ^ { 2 } = 100$$

Solve for x

$$x = \frac{\sqrt{130} - \sqrt{70}}{2} \approx 1.517576993$$

$$x=\frac{\sqrt{70}-\sqrt{130}}{2}\approx -1.517576993$$

$$x=\frac{-\sqrt{70}-\sqrt{130}}{2}\approx -9.884177258$$

$$x = \frac{\sqrt{70} + \sqrt{130}}{2} \approx 9.884177258$$

Steps Using the Quadratic Formula

To raise \frac{15}{x} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}+\frac{15^{2}}{x^{2}}=100$$

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

$$\frac{x^{2}x^{2}}{x^{2}}+\frac{15^{2}}{x^{2}}=100$$

Since \frac{x^{2}x^{2}}{x^{2}} and \frac{15^{2}}{x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}x^{2}+15^{2}}{x^{2}}=100$$

Do the multiplications in x^{2}x^{2}+15^{2}.

$$\frac{x^{4}+15^{2}}{x^{2}}=100$$

Combine like terms in x^{4}+15^{2}.

$$\frac{x^{4}+225}{x^{2}}=100$$

Subtract 100 from both sides.

$$\frac{x^{4}+225}{x^{2}}-100=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 100 times \frac{x^{2}}{x^{2}}.

$$\frac{x^{4}+225}{x^{2}}-\frac{100x^{2}}{x^{2}}=0$$

Since \frac{x^{4}+225}{x^{2}} and \frac{100x^{2}}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{4}+225-100x^{2}}{x^{2}}=0$$

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

$$x^{4}+225-100x^{2}=0$$

Substitute t for x^{2}.

$$t^{2}-100t+225=0$$

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, -100 for b, and 225 for c in the quadratic formula.

$$t=\frac{-\left(-100\right)±\sqrt{\left(-100\right)^{2}-4\times 1\times 225}}{2}$$

Do the calculations.

$$t=\frac{100±10\sqrt{91}}{2}$$

Solve the equation t=\frac{100±10\sqrt{91}}{2} when ± is plus and when ± is minus.

$$t=5\sqrt{91}+50 t=50-5\sqrt{91}$$

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

$$x=\frac{\sqrt{70}+\sqrt{130}}{2} x=-\frac{\sqrt{70}+\sqrt{130}}{2} x=-\frac{\sqrt{70}-\sqrt{130}}{2} x=\frac{\sqrt{70}-\sqrt{130}}{2}$$