Solve -x+2+x-3/x-2=0

$$- x + 2 + \frac { x - 3 } { x - 2 } = 0$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{3}i+5}{2}\approx 2.5-0.866025404i$$

$$x=\frac{5+\sqrt{3}i}{2}\approx 2.5+0.866025404i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

$$-\left(x-2\right)x+\left(x-2\right)\times 2+x-3=0$$

Use the distributive property to multiply x-2 by x.

$$-\left(x^{2}-2x\right)+\left(x-2\right)\times 2+x-3=0$$

To find the opposite of x^{2}-2x, find the opposite of each term.

$$-x^{2}+2x+\left(x-2\right)\times 2+x-3=0$$

Use the distributive property to multiply x-2 by 2.

$$-x^{2}+2x+2x-4+x-3=0$$

Combine 2x and 2x to get 4x.

$$-x^{2}+4x-4+x-3=0$$

Combine 4x and x to get 5x.

$$-x^{2}+5x-4-3=0$$

Subtract 3 from -4 to get -7.

$$-x^{2}+5x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 5 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-5±\sqrt{25+4\left(-7\right)}}{2\left(-1\right)}$$

Multiply 4 times -7.

$$x=\frac{-5±\sqrt{25-28}}{2\left(-1\right)}$$

Add 25 to -28.

$$x=\frac{-5±\sqrt{-3}}{2\left(-1\right)}$$

Take the square root of -3.

$$x=\frac{-5±\sqrt{3}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-5±\sqrt{3}i}{-2}$$

Now solve the equation x=\frac{-5±\sqrt{3}i}{-2} when ± is plus. Add -5 to i\sqrt{3}.

$$x=\frac{-5+\sqrt{3}i}{-2}$$

Divide -5+i\sqrt{3} by -2.

$$x=\frac{-\sqrt{3}i+5}{2}$$

Now solve the equation x=\frac{-5±\sqrt{3}i}{-2} when ± is minus. Subtract i\sqrt{3} from -5.

$$x=\frac{-\sqrt{3}i-5}{-2}$$

Divide -5-i\sqrt{3} by -2.

$$x=\frac{5+\sqrt{3}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{3}i+5}{2} x=\frac{5+\sqrt{3}i}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

$$-\left(x-2\right)x+\left(x-2\right)\times 2+x-3=0$$

Use the distributive property to multiply x-2 by x.

$$-\left(x^{2}-2x\right)+\left(x-2\right)\times 2+x-3=0$$

To find the opposite of x^{2}-2x, find the opposite of each term.

$$-x^{2}+2x+\left(x-2\right)\times 2+x-3=0$$

Use the distributive property to multiply x-2 by 2.

$$-x^{2}+2x+2x-4+x-3=0$$

Combine 2x and 2x to get 4x.

$$-x^{2}+4x-4+x-3=0$$

Combine 4x and x to get 5x.

$$-x^{2}+5x-4-3=0$$

Subtract 3 from -4 to get -7.

$$-x^{2}+5x-7=0$$

Add 7 to both sides. Anything plus zero gives itself.

$$-x^{2}+5x=7$$

Divide both sides by -1.

$$\frac{-x^{2}+5x}{-1}=\frac{7}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{5}{-1}x=\frac{7}{-1}$$

Divide 5 by -1.

$$x^{2}-5x=\frac{7}{-1}$$

Divide 7 by -1.

$$x^{2}-5x=-7$$

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

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-7+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-7+\frac{25}{4}$$

Add -7 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=-\frac{3}{4}$$

Factor x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=-\frac{3}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{3}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{3}i}{2} x-\frac{5}{2}=-\frac{\sqrt{3}i}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{5+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i+5}{2}$$