Solve x-2/x-5=2*+1/x-1

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

Solve for x (complex solution)

$$x=\frac{5+\sqrt{23}i}{2}\approx 2.5+2.397915762i$$

$$x=\frac{-\sqrt{23}i+5}{2}\approx 2.5-2.397915762i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 1,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x-1\right), the least common multiple of x-5,x-1.

$$\left(x-1\right)\left(x-2\right)=\left(x-5\right)\times 2\times 1$$

Use the distributive property to multiply x-1 by x-2 and combine like terms.

$$x^{2}-3x+2=\left(x-5\right)\times 2\times 1$$

Multiply 2 and 1 to get 2.

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

Use the distributive property to multiply x-5 by 2.

$$x^{2}-3x+2=2x-10$$

Subtract 2x from both sides.

$$x^{2}-3x+2-2x=-10$$

Combine -3x and -2x to get -5x.

$$x^{2}-5x+2=-10$$

Add 10 to both sides.

$$x^{2}-5x+2+10=0$$

Add 2 and 10 to get 12.

$$x^{2}-5x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 12}}{2}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 12}}{2}$$

Multiply -4 times 12.

$$x=\frac{-\left(-5\right)±\sqrt{25-48}}{2}$$

Add 25 to -48.

$$x=\frac{-\left(-5\right)±\sqrt{-23}}{2}$$

Take the square root of -23.

$$x=\frac{-\left(-5\right)±\sqrt{23}i}{2}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{23}i}{2}$$

Now solve the equation x=\frac{5±\sqrt{23}i}{2} when ± is plus. Add 5 to i\sqrt{23}.

$$x=\frac{5+\sqrt{23}i}{2}$$

Now solve the equation x=\frac{5±\sqrt{23}i}{2} when ± is minus. Subtract i\sqrt{23} from 5.

$$x=\frac{-\sqrt{23}i+5}{2}$$

The equation is now solved.

$$x=\frac{5+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+5}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 1,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x-1\right), the least common multiple of x-5,x-1.

$$\left(x-1\right)\left(x-2\right)=\left(x-5\right)\times 2\times 1$$

Use the distributive property to multiply x-1 by x-2 and combine like terms.

$$x^{2}-3x+2=\left(x-5\right)\times 2\times 1$$

Multiply 2 and 1 to get 2.

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

Use the distributive property to multiply x-5 by 2.

$$x^{2}-3x+2=2x-10$$

Subtract 2x from both sides.

$$x^{2}-3x+2-2x=-10$$

Combine -3x and -2x to get -5x.

$$x^{2}-5x+2=-10$$

Subtract 2 from both sides.

$$x^{2}-5x=-10-2$$

Subtract 2 from -10 to get -12.

$$x^{2}-5x=-12$$

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.

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

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

$$x^{2}-5x+\frac{25}{4}=-12+\frac{25}{4}$$

Add -12 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=-\frac{23}{4}$$

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}.

$$\left(x-\frac{5}{2}\right)^{2}=-\frac{23}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{23}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{23}i}{2} x-\frac{5}{2}=-\frac{\sqrt{23}i}{2}$$

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

$$x=\frac{5+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+5}{2}$$