Solve (x+4/x)^2=3x^2+4

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

Solve for x (complex solution)

$$x=\sqrt{2}i\approx 1.414213562i$$

$$x=2$$

$$x=-\sqrt{2}i\approx -0-1.414213562i$$

$$x=-2$$

Steps Using the Quadratic Formula

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

$$\left(\frac{xx}{x}+\frac{4}{x}\right)^{2}=3x^{2}+4$$

Since \frac{xx}{x} and \frac{4}{x} have the same denominator, add them by adding their numerators.

$$\left(\frac{xx+4}{x}\right)^{2}=3x^{2}+4$$

Do the multiplications in xx+4.

$$\left(\frac{x^{2}+4}{x}\right)^{2}=3x^{2}+4$$

To raise \frac{x^{2}+4}{x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(x^{2}+4\right)^{2}}{x^{2}}=3x^{2}+4$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+4\right)^{2}.

$$\frac{\left(x^{2}\right)^{2}+8x^{2}+16}{x^{2}}=3x^{2}+4$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{x^{4}+8x^{2}+16}{x^{2}}=3x^{2}+4$$

Subtract 3x^{2} from both sides.

$$\frac{x^{4}+8x^{2}+16}{x^{2}}-3x^{2}=4$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -3x^{2} times \frac{x^{2}}{x^{2}}.

$$\frac{x^{4}+8x^{2}+16}{x^{2}}+\frac{-3x^{2}x^{2}}{x^{2}}=4$$

Since \frac{x^{4}+8x^{2}+16}{x^{2}} and \frac{-3x^{2}x^{2}}{x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{4}+8x^{2}+16-3x^{2}x^{2}}{x^{2}}=4$$

Do the multiplications in x^{4}+8x^{2}+16-3x^{2}x^{2}.

$$\frac{x^{4}+8x^{2}+16-3x^{4}}{x^{2}}=4$$

Combine like terms in x^{4}+8x^{2}+16-3x^{4}.

$$\frac{-2x^{4}+8x^{2}+16}{x^{2}}=4$$

Subtract 4 from both sides.

$$\frac{-2x^{4}+8x^{2}+16}{x^{2}}-4=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{x^{2}}{x^{2}}.

$$\frac{-2x^{4}+8x^{2}+16}{x^{2}}-\frac{4x^{2}}{x^{2}}=0$$

Since \frac{-2x^{4}+8x^{2}+16}{x^{2}} and \frac{4x^{2}}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2x^{4}+8x^{2}+16-4x^{2}}{x^{2}}=0$$

Combine like terms in -2x^{4}+8x^{2}+16-4x^{2}.

$$\frac{-2x^{4}+4x^{2}+16}{x^{2}}=0$$

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

$$-2x^{4}+4x^{2}+16=0$$

Substitute t for x^{2}.

$$-2t^{2}+4t+16=0$$

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 -2 for a, 4 for b, and 16 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-4±12}{-4}$$

Solve the equation t=\frac{-4±12}{-4} when ± is plus and when ± is minus.

$$t=-2 t=4$$

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

$$x=-\sqrt{2}i x=\sqrt{2}i x=-2 x=2$$

Solve for x

$$x=-2$$

$$x=2$$

Steps Using the Quadratic Formula

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

$$\left(\frac{xx}{x}+\frac{4}{x}\right)^{2}=3x^{2}+4$$

Since \frac{xx}{x} and \frac{4}{x} have the same denominator, add them by adding their numerators.

$$\left(\frac{xx+4}{x}\right)^{2}=3x^{2}+4$$

Do the multiplications in xx+4.

$$\left(\frac{x^{2}+4}{x}\right)^{2}=3x^{2}+4$$

To raise \frac{x^{2}+4}{x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(x^{2}+4\right)^{2}}{x^{2}}=3x^{2}+4$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+4\right)^{2}.

$$\frac{\left(x^{2}\right)^{2}+8x^{2}+16}{x^{2}}=3x^{2}+4$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{x^{4}+8x^{2}+16}{x^{2}}=3x^{2}+4$$

Subtract 3x^{2} from both sides.

$$\frac{x^{4}+8x^{2}+16}{x^{2}}-3x^{2}=4$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -3x^{2} times \frac{x^{2}}{x^{2}}.

$$\frac{x^{4}+8x^{2}+16}{x^{2}}+\frac{-3x^{2}x^{2}}{x^{2}}=4$$

Since \frac{x^{4}+8x^{2}+16}{x^{2}} and \frac{-3x^{2}x^{2}}{x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{4}+8x^{2}+16-3x^{2}x^{2}}{x^{2}}=4$$

Do the multiplications in x^{4}+8x^{2}+16-3x^{2}x^{2}.

$$\frac{x^{4}+8x^{2}+16-3x^{4}}{x^{2}}=4$$

Combine like terms in x^{4}+8x^{2}+16-3x^{4}.

$$\frac{-2x^{4}+8x^{2}+16}{x^{2}}=4$$

Subtract 4 from both sides.

$$\frac{-2x^{4}+8x^{2}+16}{x^{2}}-4=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{x^{2}}{x^{2}}.

$$\frac{-2x^{4}+8x^{2}+16}{x^{2}}-\frac{4x^{2}}{x^{2}}=0$$

Since \frac{-2x^{4}+8x^{2}+16}{x^{2}} and \frac{4x^{2}}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2x^{4}+8x^{2}+16-4x^{2}}{x^{2}}=0$$

Combine like terms in -2x^{4}+8x^{2}+16-4x^{2}.

$$\frac{-2x^{4}+4x^{2}+16}{x^{2}}=0$$

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

$$-2x^{4}+4x^{2}+16=0$$

Substitute t for x^{2}.

$$-2t^{2}+4t+16=0$$

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 -2 for a, 4 for b, and 16 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-4±12}{-4}$$

Solve the equation t=\frac{-4±12}{-4} when ± is plus and when ± is minus.

$$t=-2 t=4$$

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

$$x=2 x=-2$$