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

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

Solve for x

$$x=-\frac{1}{3}\approx -0.333333333$$

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, the least common multiple of x-2,2-x.

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

Use the distributive property to multiply 3x by x-2.

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

To find the opposite of x-1, find the opposite of each term.

$$3x^{2}-6x-1=-x+1$$

Add x to both sides.

$$3x^{2}-6x-1+x=1$$

Combine -6x and x to get -5x.

$$3x^{2}-5x-1=1$$

Subtract 1 from both sides.

$$3x^{2}-5x-1-1=0$$

Subtract 1 from -1 to get -2.

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

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

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

Square -5.

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

Multiply -4 times 3.

$$x=\frac{-\left(-5\right)±\sqrt{25-12\left(-2\right)}}{2\times 3}$$

Multiply -12 times -2.

$$x=\frac{-\left(-5\right)±\sqrt{25+24}}{2\times 3}$$

Add 25 to 24.

$$x=\frac{-\left(-5\right)±\sqrt{49}}{2\times 3}$$

Take the square root of 49.

$$x=\frac{-\left(-5\right)±7}{2\times 3}$$

The opposite of -5 is 5.

$$x=\frac{5±7}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{5±7}{6}$$

Now solve the equation x=\frac{5±7}{6} when ± is plus. Add 5 to 7.

$$x=\frac{12}{6}$$

Divide 12 by 6.

$$x=2$$

Now solve the equation x=\frac{5±7}{6} when ± is minus. Subtract 7 from 5.

$$x=-\frac{2}{6}$$

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

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

The equation is now solved.

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

Variable x cannot be equal to 2.

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

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, the least common multiple of x-2,2-x.

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

Use the distributive property to multiply 3x by x-2.

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

To find the opposite of x-1, find the opposite of each term.

$$3x^{2}-6x-1=-x+1$$

Add x to both sides.

$$3x^{2}-6x-1+x=1$$

Combine -6x and x to get -5x.

$$3x^{2}-5x-1=1$$

Add 1 to both sides.

$$3x^{2}-5x=1+1$$

Add 1 and 1 to get 2.

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

Divide both sides by 3.

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

Dividing by 3 undoes the multiplication by 3.

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

Divide -\frac{5}{3}, the coefficient of the x term, by 2 to get -\frac{5}{6}. Then add the square of -\frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{2}{3}+\frac{25}{36}$$

Add \frac{2}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{49}{36}$$

Factor x^{2}-\frac{5}{3}x+\frac{25}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{6}\right)^{2}=\frac{49}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Simplify.

$$x-\frac{5}{6}=\frac{7}{6} x-\frac{5}{6}=-\frac{7}{6}$$

Add \frac{5}{6} to both sides of the equation.

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

Variable x cannot be equal to 2.

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