Solve -2x+5/2-x=2/x

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

Solve for x (complex solution)

$$x=\frac{5+\sqrt{71}i}{12}\approx 0.416666667+0.702179148i$$

$$x=\frac{-\sqrt{71}i+5}{12}\approx 0.416666667-0.702179148i$$

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.

$$-2x\times 2x+2x\times \frac{5}{2}-x\times 2x=2\times 2$$

Multiply -2 and 2 to get -4.

$$-4xx+2x\times \frac{5}{2}-x\times 2x=2\times 2$$

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

$$-4x^{2}+2x\times \frac{5}{2}-x\times 2x=2\times 2$$

Cancel out 2 and 2.

$$-4x^{2}+5x-x\times 2x=2\times 2$$

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

$$-4x^{2}+5x-x^{2}\times 2=2\times 2$$

Multiply -1 and 2 to get -2.

$$-4x^{2}+5x-2x^{2}=2\times 2$$

Combine -4x^{2} and -2x^{2} to get -6x^{2}.

$$-6x^{2}+5x=2\times 2$$

Multiply 2 and 2 to get 4.

$$-6x^{2}+5x=4$$

Subtract 4 from both sides.

$$-6x^{2}+5x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 5 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-6\right)\left(-4\right)}}{2\left(-6\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-6\right)\left(-4\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-5±\sqrt{25+24\left(-4\right)}}{2\left(-6\right)}$$

Multiply 24 times -4.

$$x=\frac{-5±\sqrt{25-96}}{2\left(-6\right)}$$

Add 25 to -96.

$$x=\frac{-5±\sqrt{-71}}{2\left(-6\right)}$$

Take the square root of -71.

$$x=\frac{-5±\sqrt{71}i}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-5±\sqrt{71}i}{-12}$$

Now solve the equation x=\frac{-5±\sqrt{71}i}{-12} when ± is plus. Add -5 to i\sqrt{71}.

$$x=\frac{-5+\sqrt{71}i}{-12}$$

Divide -5+i\sqrt{71} by -12.

$$x=\frac{-\sqrt{71}i+5}{12}$$

Now solve the equation x=\frac{-5±\sqrt{71}i}{-12} when ± is minus. Subtract i\sqrt{71} from -5.

$$x=\frac{-\sqrt{71}i-5}{-12}$$

Divide -5-i\sqrt{71} by -12.

$$x=\frac{5+\sqrt{71}i}{12}$$

The equation is now solved.

$$x=\frac{-\sqrt{71}i+5}{12} x=\frac{5+\sqrt{71}i}{12}$$

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.

$$-2x\times 2x+2x\times \frac{5}{2}-x\times 2x=2\times 2$$

Multiply -2 and 2 to get -4.

$$-4xx+2x\times \frac{5}{2}-x\times 2x=2\times 2$$

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

$$-4x^{2}+2x\times \frac{5}{2}-x\times 2x=2\times 2$$

Cancel out 2 and 2.

$$-4x^{2}+5x-x\times 2x=2\times 2$$

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

$$-4x^{2}+5x-x^{2}\times 2=2\times 2$$

Multiply -1 and 2 to get -2.

$$-4x^{2}+5x-2x^{2}=2\times 2$$

Combine -4x^{2} and -2x^{2} to get -6x^{2}.

$$-6x^{2}+5x=2\times 2$$

Multiply 2 and 2 to get 4.

$$-6x^{2}+5x=4$$

Divide both sides by -6.

$$\frac{-6x^{2}+5x}{-6}=\frac{4}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{5}{-6}x=\frac{4}{-6}$$

Divide 5 by -6.

$$x^{2}-\frac{5}{6}x=\frac{4}{-6}$$

Reduce the fraction \frac{4}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{5}{6}x=-\frac{2}{3}$$

Divide -\frac{5}{6}, the coefficient of the x term, by 2 to get -\frac{5}{12}. Then add the square of -\frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{6}x+\left(-\frac{5}{12}\right)^{2}=-\frac{2}{3}+\left(-\frac{5}{12}\right)^{2}$$

Square -\frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=-\frac{2}{3}+\frac{25}{144}$$

Add -\frac{2}{3} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=-\frac{71}{144}$$

Factor x^{2}-\frac{5}{6}x+\frac{25}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{12}\right)^{2}=-\frac{71}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{-\frac{71}{144}}$$

Simplify.

$$x-\frac{5}{12}=\frac{\sqrt{71}i}{12} x-\frac{5}{12}=-\frac{\sqrt{71}i}{12}$$

Add \frac{5}{12} to both sides of the equation.

$$x=\frac{5+\sqrt{71}i}{12} x=\frac{-\sqrt{71}i+5}{12}$$