Solve -1/2x+2=-6/x

$$- \frac { 1 } { 2 } x + 2 = - \frac { 6 } { x }$$

Solve for x

$$x=-2$$

$$x=6$$

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 2x, the least common multiple of 2,x.

$$-\frac{1}{2}x\times 2x+2x\times 2=-2\times 6$$

Cancel out 2 and 2.

$$-xx+2x\times 2=-2\times 6$$

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

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

Multiply 2 and 2 to get 4.

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

Multiply -2 and 6 to get -12.

$$-x^{2}+4x=-12$$

Add 12 to both sides.

$$-x^{2}+4x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4\times 12}}{2\left(-1\right)}$$

Multiply 4 times 12.

$$x=\frac{-4±\sqrt{16+48}}{2\left(-1\right)}$$

Add 16 to 48.

$$x=\frac{-4±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

$$x=\frac{-4±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4±8}{-2}$$

Now solve the equation x=\frac{-4±8}{-2} when ± is plus. Add -4 to 8.

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

Divide 4 by -2.

$$x=-2$$

Now solve the equation x=\frac{-4±8}{-2} when ± is minus. Subtract 8 from -4.

$$x=-\frac{12}{-2}$$

Divide -12 by -2.

$$x=6$$

The equation is now solved.

$$x=-2 x=6$$

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 2x, the least common multiple of 2,x.

$$-\frac{1}{2}x\times 2x+2x\times 2=-2\times 6$$

Cancel out 2 and 2.

$$-xx+2x\times 2=-2\times 6$$

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

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

Multiply 2 and 2 to get 4.

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

Multiply -2 and 6 to get -12.

$$-x^{2}+4x=-12$$

Divide both sides by -1.

$$\frac{-x^{2}+4x}{-1}=-\frac{12}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4}{-1}x=-\frac{12}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=-\frac{12}{-1}$$

Divide -12 by -1.

$$x^{2}-4x=12$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=12+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=12+4$$

Add 12 to 4.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-2=4 x-2=-4$$

Add 2 to both sides of the equation.

$$x=6 x=-2$$