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

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

Solve for x (complex solution)

$$x=-\frac{\sqrt{6}i}{3}\approx -0-0.816496581i$$

$$x=\frac{\sqrt{6}i}{3}\approx 0.816496581i$$

$$x=1$$

$$x=-1$$

Steps Using the Quadratic Formula

Combine 4x and x to get 5x.

$$\left(5x\right)^{2}=\left(x+\frac{4}{x}\right)^{2}$$

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}=\left(x+\frac{4}{x}\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}=\left(x+\frac{4}{x}\right)^{2}$$

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

$$25x^{2}=\left(\frac{xx}{x}+\frac{4}{x}\right)^{2}$$

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

$$25x^{2}=\left(\frac{xx+4}{x}\right)^{2}$$

Do the multiplications in xx+4.

$$25x^{2}=\left(\frac{x^{2}+4}{x}\right)^{2}$$

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

$$25x^{2}=\frac{\left(x^{2}+4\right)^{2}}{x^{2}}$$

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

$$25x^{2}=\frac{\left(x^{2}\right)^{2}+8x^{2}+16}{x^{2}}$$

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

$$25x^{2}=\frac{x^{4}+8x^{2}+16}{x^{2}}$$

Subtract \frac{x^{4}+8x^{2}+16}{x^{2}} from both sides.

$$25x^{2}-\frac{x^{4}+8x^{2}+16}{x^{2}}=0$$

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

$$\frac{25x^{2}x^{2}}{x^{2}}-\frac{x^{4}+8x^{2}+16}{x^{2}}=0$$

Since \frac{25x^{2}x^{2}}{x^{2}} and \frac{x^{4}+8x^{2}+16}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{25x^{2}x^{2}-\left(x^{4}+8x^{2}+16\right)}{x^{2}}=0$$

Do the multiplications in 25x^{2}x^{2}-\left(x^{4}+8x^{2}+16\right).

$$\frac{25x^{4}-x^{4}-8x^{2}-16}{x^{2}}=0$$

Combine like terms in 25x^{4}-x^{4}-8x^{2}-16.

$$\frac{24x^{4}-8x^{2}-16}{x^{2}}=0$$

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

$$24x^{4}-8x^{2}-16=0$$

Substitute t for x^{2}.

$$24t^{2}-8t-16=0$$

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 24 for a, -8 for b, and -16 for c in the quadratic formula.

$$t=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 24\left(-16\right)}}{2\times 24}$$

Do the calculations.

$$t=\frac{8±40}{48}$$

Solve the equation t=\frac{8±40}{48} when ± is plus and when ± is minus.

$$t=1 t=-\frac{2}{3}$$

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

$$x=-1 x=1 x=-\frac{\sqrt{6}i}{3} x=\frac{\sqrt{6}i}{3}$$

Solve for x

$$x=-1$$

$$x=1$$

Steps Using the Quadratic Formula

Combine 4x and x to get 5x.

$$\left(5x\right)^{2}=\left(x+\frac{4}{x}\right)^{2}$$

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}=\left(x+\frac{4}{x}\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}=\left(x+\frac{4}{x}\right)^{2}$$

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

$$25x^{2}=\left(\frac{xx}{x}+\frac{4}{x}\right)^{2}$$

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

$$25x^{2}=\left(\frac{xx+4}{x}\right)^{2}$$

Do the multiplications in xx+4.

$$25x^{2}=\left(\frac{x^{2}+4}{x}\right)^{2}$$

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

$$25x^{2}=\frac{\left(x^{2}+4\right)^{2}}{x^{2}}$$

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

$$25x^{2}=\frac{\left(x^{2}\right)^{2}+8x^{2}+16}{x^{2}}$$

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

$$25x^{2}=\frac{x^{4}+8x^{2}+16}{x^{2}}$$

Subtract \frac{x^{4}+8x^{2}+16}{x^{2}} from both sides.

$$25x^{2}-\frac{x^{4}+8x^{2}+16}{x^{2}}=0$$

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

$$\frac{25x^{2}x^{2}}{x^{2}}-\frac{x^{4}+8x^{2}+16}{x^{2}}=0$$

Since \frac{25x^{2}x^{2}}{x^{2}} and \frac{x^{4}+8x^{2}+16}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{25x^{2}x^{2}-\left(x^{4}+8x^{2}+16\right)}{x^{2}}=0$$

Do the multiplications in 25x^{2}x^{2}-\left(x^{4}+8x^{2}+16\right).

$$\frac{25x^{4}-x^{4}-8x^{2}-16}{x^{2}}=0$$

Combine like terms in 25x^{4}-x^{4}-8x^{2}-16.

$$\frac{24x^{4}-8x^{2}-16}{x^{2}}=0$$

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

$$24x^{4}-8x^{2}-16=0$$

Substitute t for x^{2}.

$$24t^{2}-8t-16=0$$

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 24 for a, -8 for b, and -16 for c in the quadratic formula.

$$t=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 24\left(-16\right)}}{2\times 24}$$

Do the calculations.

$$t=\frac{8±40}{48}$$

Solve the equation t=\frac{8±40}{48} when ± is plus and when ± is minus.

$$t=1 t=-\frac{2}{3}$$

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

$$x=1 x=-1$$