Solve x+4+6/x=0

$$x+4+ \frac{ 6 }{ x } = 0$$

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.

$$xx+x\times 4+6=0$$

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

$$x^{2}+x\times 4+6=0$$

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}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 6}}{2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 6}}{2}$$

Multiply -4 times 6.

$$x=\frac{-4±\sqrt{16-24}}{2}$$

Add 16 to -24.

$$x=\frac{-4±\sqrt{-8}}{2}$$

Take the square root of -8.

$$x=\frac{-4±2\sqrt{2}i}{2}$$

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.

$$xx+x\times 4+6=0$$

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

$$x^{2}+x\times 4+6=0$$

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

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

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

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.

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

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.

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

Subtract 2 from both sides of the equation.

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