Solve {left(frac{6sqrt{11}}{x}right)}^2+36=1/4x^2

$${ \left( \frac{ 6 \sqrt{ 11 } }{ x } \right) }^{ 2 } +36= \frac{ 1 }{ 4 } { x }^{ 2 }$$

Solve for x (complex solution)

$$x=-2i\sqrt{3\left(\sqrt{47}-6\right)}\approx -0-3.20434942i$$

$$x=2i\sqrt{3\left(\sqrt{47}-6\right)}\approx 3.20434942i$$

$$x=2\sqrt{3\left(\sqrt{47}+6\right)}\approx 12.420461151$$

$$x=-2\sqrt{3\left(\sqrt{47}+6\right)}\approx -12.420461151$$

Steps Using the Quadratic Formula

To raise \frac{6\sqrt{11}}{x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(6\sqrt{11}\right)^{2}}{x^{2}}+36=\frac{1}{4}x^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 36 times \frac{x^{2}}{x^{2}}.

$$\frac{\left(6\sqrt{11}\right)^{2}}{x^{2}}+\frac{36x^{2}}{x^{2}}=\frac{1}{4}x^{2}$$

Since \frac{\left(6\sqrt{11}\right)^{2}}{x^{2}} and \frac{36x^{2}}{x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(6\sqrt{11}\right)^{2}+36x^{2}}{x^{2}}=\frac{1}{4}x^{2}$$

Expand \left(6\sqrt{11}\right)^{2}.

$$\frac{6^{2}\left(\sqrt{11}\right)^{2}+36x^{2}}{x^{2}}=\frac{1}{4}x^{2}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{36\left(\sqrt{11}\right)^{2}+36x^{2}}{x^{2}}=\frac{1}{4}x^{2}$$

The square of \sqrt{11} is 11.

$$\frac{36\times 11+36x^{2}}{x^{2}}=\frac{1}{4}x^{2}$$

Multiply 36 and 11 to get 396.

$$\frac{396+36x^{2}}{x^{2}}=\frac{1}{4}x^{2}$$

Subtract \frac{1}{4}x^{2} from both sides.

$$\frac{396+36x^{2}}{x^{2}}-\frac{1}{4}x^{2}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x^{2}, the least common multiple of x^{2},4.

$$4\left(396+36x^{2}\right)-\frac{1}{4}x^{2}\times 4x^{2}=0$$

Reorder the terms.

$$-\frac{1}{4}\times 4x^{2}x^{2}+4\left(36x^{2}+396\right)=0$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$-\frac{1}{4}\times 4x^{4}+4\left(36x^{2}+396\right)=0$$

Multiply -\frac{1}{4} and 4 to get -1.

$$-x^{4}+4\left(36x^{2}+396\right)=0$$

Use the distributive property to multiply 4 by 36x^{2}+396.

$$-x^{4}+144x^{2}+1584=0$$

Substitute t for x^{2}.

$$-t^{2}+144t+1584=0$$

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, 144 for b, and 1584 for c in the quadratic formula.

$$t=\frac{-144±\sqrt{144^{2}-4\left(-1\right)\times 1584}}{-2}$$

Do the calculations.

$$t=\frac{-144±24\sqrt{47}}{-2}$$

Solve the equation t=\frac{-144±24\sqrt{47}}{-2} when ± is plus and when ± is minus.

$$t=72-12\sqrt{47} t=12\sqrt{47}+72$$

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

$$x=-i\sqrt{-\left(72-12\sqrt{47}\right)} x=i\sqrt{-\left(72-12\sqrt{47}\right)} x=-\sqrt{12\sqrt{47}+72} x=\sqrt{12\sqrt{47}+72}$$

Solve for x

$$x=2\sqrt{3\left(\sqrt{47}+6\right)}\approx 12.420461151$$

$$x=-2\sqrt{3\left(\sqrt{47}+6\right)}\approx -12.420461151$$

Steps Using the Quadratic Formula

To raise \frac{6\sqrt{11}}{x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(6\sqrt{11}\right)^{2}}{x^{2}}+36=\frac{1}{4}x^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 36 times \frac{x^{2}}{x^{2}}.

$$\frac{\left(6\sqrt{11}\right)^{2}}{x^{2}}+\frac{36x^{2}}{x^{2}}=\frac{1}{4}x^{2}$$

Since \frac{\left(6\sqrt{11}\right)^{2}}{x^{2}} and \frac{36x^{2}}{x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(6\sqrt{11}\right)^{2}+36x^{2}}{x^{2}}=\frac{1}{4}x^{2}$$

Expand \left(6\sqrt{11}\right)^{2}.

$$\frac{6^{2}\left(\sqrt{11}\right)^{2}+36x^{2}}{x^{2}}=\frac{1}{4}x^{2}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{36\left(\sqrt{11}\right)^{2}+36x^{2}}{x^{2}}=\frac{1}{4}x^{2}$$

The square of \sqrt{11} is 11.

$$\frac{36\times 11+36x^{2}}{x^{2}}=\frac{1}{4}x^{2}$$

Multiply 36 and 11 to get 396.

$$\frac{396+36x^{2}}{x^{2}}=\frac{1}{4}x^{2}$$

Subtract \frac{1}{4}x^{2} from both sides.

$$\frac{396+36x^{2}}{x^{2}}-\frac{1}{4}x^{2}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x^{2}, the least common multiple of x^{2},4.

$$4\left(396+36x^{2}\right)-\frac{1}{4}x^{2}\times 4x^{2}=0$$

Reorder the terms.

$$-\frac{1}{4}\times 4x^{2}x^{2}+4\left(36x^{2}+396\right)=0$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$-\frac{1}{4}\times 4x^{4}+4\left(36x^{2}+396\right)=0$$

Multiply -\frac{1}{4} and 4 to get -1.

$$-x^{4}+4\left(36x^{2}+396\right)=0$$

Use the distributive property to multiply 4 by 36x^{2}+396.

$$-x^{4}+144x^{2}+1584=0$$

Substitute t for x^{2}.

$$-t^{2}+144t+1584=0$$

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, 144 for b, and 1584 for c in the quadratic formula.

$$t=\frac{-144±\sqrt{144^{2}-4\left(-1\right)\times 1584}}{-2}$$

Do the calculations.

$$t=\frac{-144±24\sqrt{47}}{-2}$$

Solve the equation t=\frac{-144±24\sqrt{47}}{-2} when ± is plus and when ± is minus.

$$t=72-12\sqrt{47} t=12\sqrt{47}+72$$

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

$$x=2\sqrt{3\sqrt{47}+18} x=-2\sqrt{3\sqrt{47}+18}$$