Solve 3=2*x+1/(x-1)(x-2)+6*1/x-2

$$3 = 2 \cdot \frac { x + 1 } { ( x - 1 ) ( x - 2 ) } + 6 \cdot \frac { 1 } { x - 2 }$$

Solve for x

$$x=\frac{2}{3}\approx 0.666666667$$

$$x=5$$

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 \left(x-1\right)\left(x-2\right),x-2.

$$3\left(x-2\right)\left(x-1\right)=2\left(x+1\right)+6\left(x-1\right)\times 1$$

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

$$\left(3x-6\right)\left(x-1\right)=2\left(x+1\right)+6\left(x-1\right)\times 1$$

Use the distributive property to multiply 3x-6 by x-1 and combine like terms.

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

Use the distributive property to multiply 2 by x+1.

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

Multiply 6 and 1 to get 6.

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

Use the distributive property to multiply 6 by x-1.

$$3x^{2}-9x+6=2x+2+6x-6$$

Combine 2x and 6x to get 8x.

$$3x^{2}-9x+6=8x+2-6$$

Subtract 6 from 2 to get -4.

$$3x^{2}-9x+6=8x-4$$

Subtract 8x from both sides.

$$3x^{2}-9x+6-8x=-4$$

Combine -9x and -8x to get -17x.

$$3x^{2}-17x+6=-4$$

Add 4 to both sides.

$$3x^{2}-17x+6+4=0$$

Add 6 and 4 to get 10.

$$3x^{2}-17x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -17 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 3\times 10}}{2\times 3}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\times 3\times 10}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-17\right)±\sqrt{289-12\times 10}}{2\times 3}$$

Multiply -12 times 10.

$$x=\frac{-\left(-17\right)±\sqrt{289-120}}{2\times 3}$$

Add 289 to -120.

$$x=\frac{-\left(-17\right)±\sqrt{169}}{2\times 3}$$

Take the square root of 169.

$$x=\frac{-\left(-17\right)±13}{2\times 3}$$

The opposite of -17 is 17.

$$x=\frac{17±13}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{17±13}{6}$$

Now solve the equation x=\frac{17±13}{6} when ± is plus. Add 17 to 13.

$$x=\frac{30}{6}$$

Divide 30 by 6.

$$x=5$$

Now solve the equation x=\frac{17±13}{6} when ± is minus. Subtract 13 from 17.

$$x=\frac{4}{6}$$

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

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

The equation is now solved.

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

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 \left(x-1\right)\left(x-2\right),x-2.

$$3\left(x-2\right)\left(x-1\right)=2\left(x+1\right)+6\left(x-1\right)\times 1$$

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

$$\left(3x-6\right)\left(x-1\right)=2\left(x+1\right)+6\left(x-1\right)\times 1$$

Use the distributive property to multiply 3x-6 by x-1 and combine like terms.

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

Use the distributive property to multiply 2 by x+1.

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

Multiply 6 and 1 to get 6.

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

Use the distributive property to multiply 6 by x-1.

$$3x^{2}-9x+6=2x+2+6x-6$$

Combine 2x and 6x to get 8x.

$$3x^{2}-9x+6=8x+2-6$$

Subtract 6 from 2 to get -4.

$$3x^{2}-9x+6=8x-4$$

Subtract 8x from both sides.

$$3x^{2}-9x+6-8x=-4$$

Combine -9x and -8x to get -17x.

$$3x^{2}-17x+6=-4$$

Subtract 6 from both sides.

$$3x^{2}-17x=-4-6$$

Subtract 6 from -4 to get -10.

$$3x^{2}-17x=-10$$

Divide both sides by 3.

$$\frac{3x^{2}-17x}{3}=-\frac{10}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{17}{3}x=-\frac{10}{3}$$

Divide -\frac{17}{3}, the coefficient of the x term, by 2 to get -\frac{17}{6}. Then add the square of -\frac{17}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{17}{3}x+\left(-\frac{17}{6}\right)^{2}=-\frac{10}{3}+\left(-\frac{17}{6}\right)^{2}$$

Square -\frac{17}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{17}{3}x+\frac{289}{36}=-\frac{10}{3}+\frac{289}{36}$$

Add -\frac{10}{3} to \frac{289}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{17}{3}x+\frac{289}{36}=\frac{169}{36}$$

Factor x^{2}-\frac{17}{3}x+\frac{289}{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}.

$$\left(x-\frac{17}{6}\right)^{2}=\frac{169}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{6}\right)^{2}}=\sqrt{\frac{169}{36}}$$

Simplify.

$$x-\frac{17}{6}=\frac{13}{6} x-\frac{17}{6}=-\frac{13}{6}$$

Add \frac{17}{6} to both sides of the equation.

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