Solve x/3x-3-2/3=x-6/x-2

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

Solve for x

$$x = \frac{\sqrt{273} + 25}{8} \approx 5.190338955$$

$$x = \frac{25 - \sqrt{273}}{8} \approx 1.059661045$$

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 3\left(x-2\right)\left(x-1\right), the least common multiple of 3x-3,3,x-2.

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

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

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

Multiply 3 and -\frac{2}{3} to get -2.

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

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

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

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

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

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

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

Combine -2x and 6x to get 4x.

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

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

$$-x^{2}+4x-4=3x^{2}-21x+18$$

Subtract 3x^{2} from both sides.

$$-x^{2}+4x-4-3x^{2}=-21x+18$$

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

$$-4x^{2}+4x-4=-21x+18$$

Add 21x to both sides.

$$-4x^{2}+4x-4+21x=18$$

Combine 4x and 21x to get 25x.

$$-4x^{2}+25x-4=18$$

Subtract 18 from both sides.

$$-4x^{2}+25x-4-18=0$$

Subtract 18 from -4 to get -22.

$$-4x^{2}+25x-22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 25 for b, and -22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\left(-4\right)\left(-22\right)}}{2\left(-4\right)}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\left(-4\right)\left(-22\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-25±\sqrt{625+16\left(-22\right)}}{2\left(-4\right)}$$

Multiply 16 times -22.

$$x=\frac{-25±\sqrt{625-352}}{2\left(-4\right)}$$

Add 625 to -352.

$$x=\frac{-25±\sqrt{273}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-25±\sqrt{273}}{-8}$$

Now solve the equation x=\frac{-25±\sqrt{273}}{-8} when ± is plus. Add -25 to \sqrt{273}.

$$x=\frac{\sqrt{273}-25}{-8}$$

Divide -25+\sqrt{273} by -8.

$$x=\frac{25-\sqrt{273}}{8}$$

Now solve the equation x=\frac{-25±\sqrt{273}}{-8} when ± is minus. Subtract \sqrt{273} from -25.

$$x=\frac{-\sqrt{273}-25}{-8}$$

Divide -25-\sqrt{273} by -8.

$$x=\frac{\sqrt{273}+25}{8}$$

The equation is now solved.

$$x=\frac{25-\sqrt{273}}{8} x=\frac{\sqrt{273}+25}{8}$$

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 3\left(x-2\right)\left(x-1\right), the least common multiple of 3x-3,3,x-2.

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

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

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

Multiply 3 and -\frac{2}{3} to get -2.

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

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

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

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

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

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

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

Combine -2x and 6x to get 4x.

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

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

$$-x^{2}+4x-4=3x^{2}-21x+18$$

Subtract 3x^{2} from both sides.

$$-x^{2}+4x-4-3x^{2}=-21x+18$$

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

$$-4x^{2}+4x-4=-21x+18$$

Add 21x to both sides.

$$-4x^{2}+4x-4+21x=18$$

Combine 4x and 21x to get 25x.

$$-4x^{2}+25x-4=18$$

Add 4 to both sides.

$$-4x^{2}+25x=18+4$$

Add 18 and 4 to get 22.

$$-4x^{2}+25x=22$$

Divide both sides by -4.

$$\frac{-4x^{2}+25x}{-4}=\frac{22}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{25}{-4}x=\frac{22}{-4}$$

Divide 25 by -4.

$$x^{2}-\frac{25}{4}x=\frac{22}{-4}$$

Reduce the fraction \frac{22}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{25}{4}x=-\frac{11}{2}$$

Divide -\frac{25}{4}, the coefficient of the x term, by 2 to get -\frac{25}{8}. Then add the square of -\frac{25}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{4}x+\left(-\frac{25}{8}\right)^{2}=-\frac{11}{2}+\left(-\frac{25}{8}\right)^{2}$$

Square -\frac{25}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{4}x+\frac{625}{64}=-\frac{11}{2}+\frac{625}{64}$$

Add -\frac{11}{2} to \frac{625}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{25}{4}x+\frac{625}{64}=\frac{273}{64}$$

Factor x^{2}-\frac{25}{4}x+\frac{625}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{8}\right)^{2}=\frac{273}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{8}\right)^{2}}=\sqrt{\frac{273}{64}}$$

Simplify.

$$x-\frac{25}{8}=\frac{\sqrt{273}}{8} x-\frac{25}{8}=-\frac{\sqrt{273}}{8}$$

Add \frac{25}{8} to both sides of the equation.

$$x=\frac{\sqrt{273}+25}{8} x=\frac{25-\sqrt{273}}{8}$$