Solve x/4-1=4/x

$$\frac { x } { 4 } - 1 = \frac { 4 } { x }$$

Solve for x

$$x=2\sqrt{5}+2\approx 6.472135955$$

$$x=2-2\sqrt{5}\approx -2.472135955$$

Steps Using the Quadratic Formula

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

$$xx+4x\left(-1\right)=4\times 4$$

Multiply x and x to get x^{2}.

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

Multiply 4 and -1 to get -4.

$$x^{2}-4x=4\times 4$$

Multiply 4 and 4 to get 16.

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

Subtract 16 from both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -4.

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

Multiply -4 times -16.

$$x=\frac{-\left(-4\right)±\sqrt{16+64}}{2}$$

Add 16 to 64.

$$x=\frac{-\left(-4\right)±\sqrt{80}}{2}$$

Take the square root of 80.

$$x=\frac{-\left(-4\right)±4\sqrt{5}}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{5}}{2}$$

Now solve the equation x=\frac{4±4\sqrt{5}}{2} when ± is plus. Add 4 to 4\sqrt{5}.

$$x=\frac{4\sqrt{5}+4}{2}$$

Divide 4+4\sqrt{5} by 2.

$$x=2\sqrt{5}+2$$

Now solve the equation x=\frac{4±4\sqrt{5}}{2} when ± is minus. Subtract 4\sqrt{5} from 4.

$$x=\frac{4-4\sqrt{5}}{2}$$

Divide 4-4\sqrt{5} by 2.

$$x=2-2\sqrt{5}$$

The equation is now solved.

$$x=2\sqrt{5}+2 x=2-2\sqrt{5}$$

Steps for Completing the Square

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

$$xx+4x\left(-1\right)=4\times 4$$

Multiply x and x to get x^{2}.

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

Multiply 4 and -1 to get -4.

$$x^{2}-4x=4\times 4$$

Multiply 4 and 4 to get 16.

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

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -2.

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

Add 16 to 4.

$$x^{2}-4x+4=20$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=20$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{20}$$

Simplify.

$$x-2=2\sqrt{5} x-2=-2\sqrt{5}$$

Add 2 to both sides of the equation.

$$x=2\sqrt{5}+2 x=2-2\sqrt{5}$$