Solve -2x+1/x-1+4x-1/x-2=-2

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

Solve for x (complex solution)

$$x=\frac{3+\sqrt{3}i}{4}\approx 0.75+0.433012702i$$

$$x=\frac{-\sqrt{3}i+3}{4}\approx 0.75-0.433012702i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right), the least common multiple of x-1,x-2.

$$\left(x-2\right)\left(-2x+1\right)+\left(x-1\right)\left(4x-1\right)=-2\left(x-2\right)\left(x-1\right)$$

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

$$-2x^{2}+5x-2+\left(x-1\right)\left(4x-1\right)=-2\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x-1 by 4x-1 and combine like terms.

$$-2x^{2}+5x-2+4x^{2}-5x+1=-2\left(x-2\right)\left(x-1\right)$$

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

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

Combine 5x and -5x to get 0.

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

Add -2 and 1 to get -1.

$$2x^{2}-1=-2\left(x-2\right)\left(x-1\right)$$

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

$$2x^{2}-1=\left(-2x+4\right)\left(x-1\right)$$

Use the distributive property to multiply -2x+4 by x-1 and combine like terms.

$$2x^{2}-1=-2x^{2}+6x-4$$

Add 2x^{2} to both sides.

$$2x^{2}-1+2x^{2}=6x-4$$

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

$$4x^{2}-1=6x-4$$

Subtract 6x from both sides.

$$4x^{2}-1-6x=-4$$

Add 4 to both sides.

$$4x^{2}-1-6x+4=0$$

Add -1 and 4 to get 3.

$$4x^{2}+3-6x=0$$

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.

$$4x^{2}-6x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -6 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 4\times 3}}{2\times 4}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 4\times 3}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-6\right)±\sqrt{36-16\times 3}}{2\times 4}$$

Multiply -16 times 3.

$$x=\frac{-\left(-6\right)±\sqrt{36-48}}{2\times 4}$$

Add 36 to -48.

$$x=\frac{-\left(-6\right)±\sqrt{-12}}{2\times 4}$$

Take the square root of -12.

$$x=\frac{-\left(-6\right)±2\sqrt{3}i}{2\times 4}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{3}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{6±2\sqrt{3}i}{8}$$

Now solve the equation x=\frac{6±2\sqrt{3}i}{8} when ± is plus. Add 6 to 2i\sqrt{3}.

$$x=\frac{6+2\sqrt{3}i}{8}$$

Divide 6+2i\sqrt{3} by 8.

$$x=\frac{3+\sqrt{3}i}{4}$$

Now solve the equation x=\frac{6±2\sqrt{3}i}{8} when ± is minus. Subtract 2i\sqrt{3} from 6.

$$x=\frac{-2\sqrt{3}i+6}{8}$$

Divide 6-2i\sqrt{3} by 8.

$$x=\frac{-\sqrt{3}i+3}{4}$$

The equation is now solved.

$$x=\frac{3+\sqrt{3}i}{4} x=\frac{-\sqrt{3}i+3}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right), the least common multiple of x-1,x-2.

$$\left(x-2\right)\left(-2x+1\right)+\left(x-1\right)\left(4x-1\right)=-2\left(x-2\right)\left(x-1\right)$$

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

$$-2x^{2}+5x-2+\left(x-1\right)\left(4x-1\right)=-2\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x-1 by 4x-1 and combine like terms.

$$-2x^{2}+5x-2+4x^{2}-5x+1=-2\left(x-2\right)\left(x-1\right)$$

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

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

Combine 5x and -5x to get 0.

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

Add -2 and 1 to get -1.

$$2x^{2}-1=-2\left(x-2\right)\left(x-1\right)$$

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

$$2x^{2}-1=\left(-2x+4\right)\left(x-1\right)$$

Use the distributive property to multiply -2x+4 by x-1 and combine like terms.

$$2x^{2}-1=-2x^{2}+6x-4$$

Add 2x^{2} to both sides.

$$2x^{2}-1+2x^{2}=6x-4$$

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

$$4x^{2}-1=6x-4$$

Subtract 6x from both sides.

$$4x^{2}-1-6x=-4$$

Add 1 to both sides.

$$4x^{2}-6x=-4+1$$

Add -4 and 1 to get -3.

$$4x^{2}-6x=-3$$

Divide both sides by 4.

$$\frac{4x^{2}-6x}{4}=-\frac{3}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{6}{4}\right)x=-\frac{3}{4}$$

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

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

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{3}{4}+\frac{9}{16}$$

Add -\frac{3}{4} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{3}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{4}\right)^{2}=-\frac{3}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{-\frac{3}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{\sqrt{3}i}{4} x-\frac{3}{4}=-\frac{\sqrt{3}i}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=\frac{3+\sqrt{3}i}{4} x=\frac{-\sqrt{3}i+3}{4}$$