Solve 3-x/3x-1-x-1/x-2=-2

$$\frac { 3 - x } { 3 x - 1 } - \frac { x - 1 } { x - 2 } = - 2$$

Solve for x

$$x=-\frac{1}{2}=-0.5$$

$$x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{1}{3},2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(3x-1\right), the least common multiple of 3x-1,x-2.

$$\left(x-2\right)\left(3-x\right)-\left(3x-1\right)\left(x-1\right)=-2\left(x-2\right)\left(3x-1\right)$$

Use the distributive property to multiply x-2 by 3-x and combine like terms.

$$5x-x^{2}-6-\left(3x-1\right)\left(x-1\right)=-2\left(x-2\right)\left(3x-1\right)$$

Use the distributive property to multiply 3x-1 by x-1 and combine like terms.

$$5x-x^{2}-6-\left(3x^{2}-4x+1\right)=-2\left(x-2\right)\left(3x-1\right)$$

To find the opposite of 3x^{2}-4x+1, find the opposite of each term.

$$5x-x^{2}-6-3x^{2}+4x-1=-2\left(x-2\right)\left(3x-1\right)$$

Combine -x^{2} and -3x^{2} to get -4x^{2}.

$$5x-4x^{2}-6+4x-1=-2\left(x-2\right)\left(3x-1\right)$$

Combine 5x and 4x to get 9x.

$$9x-4x^{2}-6-1=-2\left(x-2\right)\left(3x-1\right)$$

Subtract 1 from -6 to get -7.

$$9x-4x^{2}-7=-2\left(x-2\right)\left(3x-1\right)$$

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

$$9x-4x^{2}-7=\left(-2x+4\right)\left(3x-1\right)$$

Use the distributive property to multiply -2x+4 by 3x-1 and combine like terms.

$$9x-4x^{2}-7=-6x^{2}+14x-4$$

Add 6x^{2} to both sides.

$$9x-4x^{2}-7+6x^{2}=14x-4$$

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

$$9x+2x^{2}-7=14x-4$$

Subtract 14x from both sides.

$$9x+2x^{2}-7-14x=-4$$

Combine 9x and -14x to get -5x.

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

Add 4 to both sides.

$$-5x+2x^{2}-7+4=0$$

Add -7 and 4 to get -3.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -5 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 2\left(-3\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-5\right)±\sqrt{25-8\left(-3\right)}}{2\times 2}$$

Multiply -8 times -3.

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

Add 25 to 24.

$$x=\frac{-\left(-5\right)±\sqrt{49}}{2\times 2}$$

Take the square root of 49.

$$x=\frac{-\left(-5\right)±7}{2\times 2}$$

The opposite of -5 is 5.

$$x=\frac{5±7}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{5±7}{4}$$

Now solve the equation x=\frac{5±7}{4} when ± is plus. Add 5 to 7.

$$x=\frac{12}{4}$$

Divide 12 by 4.

$$x=3$$

Now solve the equation x=\frac{5±7}{4} when ± is minus. Subtract 7 from 5.

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

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

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

The equation is now solved.

$$x=3 x=-\frac{1}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{1}{3},2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(3x-1\right), the least common multiple of 3x-1,x-2.

$$\left(x-2\right)\left(3-x\right)-\left(3x-1\right)\left(x-1\right)=-2\left(x-2\right)\left(3x-1\right)$$

Use the distributive property to multiply x-2 by 3-x and combine like terms.

$$5x-x^{2}-6-\left(3x-1\right)\left(x-1\right)=-2\left(x-2\right)\left(3x-1\right)$$

Use the distributive property to multiply 3x-1 by x-1 and combine like terms.

$$5x-x^{2}-6-\left(3x^{2}-4x+1\right)=-2\left(x-2\right)\left(3x-1\right)$$

To find the opposite of 3x^{2}-4x+1, find the opposite of each term.

$$5x-x^{2}-6-3x^{2}+4x-1=-2\left(x-2\right)\left(3x-1\right)$$

Combine -x^{2} and -3x^{2} to get -4x^{2}.

$$5x-4x^{2}-6+4x-1=-2\left(x-2\right)\left(3x-1\right)$$

Combine 5x and 4x to get 9x.

$$9x-4x^{2}-6-1=-2\left(x-2\right)\left(3x-1\right)$$

Subtract 1 from -6 to get -7.

$$9x-4x^{2}-7=-2\left(x-2\right)\left(3x-1\right)$$

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

$$9x-4x^{2}-7=\left(-2x+4\right)\left(3x-1\right)$$

Use the distributive property to multiply -2x+4 by 3x-1 and combine like terms.

$$9x-4x^{2}-7=-6x^{2}+14x-4$$

Add 6x^{2} to both sides.

$$9x-4x^{2}-7+6x^{2}=14x-4$$

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

$$9x+2x^{2}-7=14x-4$$

Subtract 14x from both sides.

$$9x+2x^{2}-7-14x=-4$$

Combine 9x and -14x to get -5x.

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

Add 7 to both sides.

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

Add -4 and 7 to get 3.

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

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$2x^{2}-5x=3$$

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{3}{2}+\frac{25}{16}$$

Add \frac{3}{2} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{49}{16}$$

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

$$\left(x-\frac{5}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x-\frac{5}{4}=\frac{7}{4} x-\frac{5}{4}=-\frac{7}{4}$$

Add \frac{5}{4} to both sides of the equation.

$$x=3 x=-\frac{1}{2}$$