Solve x+6/x=4

$$x+ \frac{ 6 }{ x } =4$$

Solve for x (complex solution)

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

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

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 x.

$$xx+6=4x$$

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

$$x^{2}+6=4x$$

Subtract 4x from both sides.

$$x^{2}+6-4x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-4x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 6}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 6}}{2}$$

Multiply -4 times 6.

$$x=\frac{-\left(-4\right)±\sqrt{16-24}}{2}$$

Add 16 to -24.

$$x=\frac{-\left(-4\right)±\sqrt{-8}}{2}$$

Take the square root of -8.

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

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{2}i}{2}$$

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

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

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

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

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

$$x=\frac{-2\sqrt{2}i+4}{2}$$

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

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

The equation is now solved.

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

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 x.

$$xx+6=4x$$

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

$$x^{2}+6=4x$$

Subtract 4x from both sides.

$$x^{2}+6-4x=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-4x=-6$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=-6+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-6+4$$

Add -6 to 4.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 2 to both sides of the equation.

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