Solve x-1/x-2+x-3/x-4=31/3

$$\frac { x - 1 } { x - 2 } + \frac { x - 3 } { x - 4 } = 3 \frac { 1 } { 3 }$$

Solve for x

$$x=5$$

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 2,4 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-4\right)\left(x-2\right), the least common multiple of x-2,x-4,3.

$$\left(3x-12\right)\left(x-1\right)+\left(3x-6\right)\left(x-3\right)=\left(x-4\right)\left(x-2\right)\left(3\times 3+1\right)$$

Use the distributive property to multiply 3x-12 by x-1 and combine like terms.

$$3x^{2}-15x+12+\left(3x-6\right)\left(x-3\right)=\left(x-4\right)\left(x-2\right)\left(3\times 3+1\right)$$

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

$$3x^{2}-15x+12+3x^{2}-15x+18=\left(x-4\right)\left(x-2\right)\left(3\times 3+1\right)$$

Combine 3x^{2} and 3x^{2} to get 6x^{2}.

$$6x^{2}-15x+12-15x+18=\left(x-4\right)\left(x-2\right)\left(3\times 3+1\right)$$

Combine -15x and -15x to get -30x.

$$6x^{2}-30x+12+18=\left(x-4\right)\left(x-2\right)\left(3\times 3+1\right)$$

Add 12 and 18 to get 30.

$$6x^{2}-30x+30=\left(x-4\right)\left(x-2\right)\left(3\times 3+1\right)$$

Multiply 3 and 3 to get 9.

$$6x^{2}-30x+30=\left(x-4\right)\left(x-2\right)\left(9+1\right)$$

Add 9 and 1 to get 10.

$$6x^{2}-30x+30=\left(x-4\right)\left(x-2\right)\times 10$$

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

$$6x^{2}-30x+30=\left(x^{2}-6x+8\right)\times 10$$

Use the distributive property to multiply x^{2}-6x+8 by 10.

$$6x^{2}-30x+30=10x^{2}-60x+80$$

Subtract 10x^{2} from both sides.

$$6x^{2}-30x+30-10x^{2}=-60x+80$$

Combine 6x^{2} and -10x^{2} to get -4x^{2}.

$$-4x^{2}-30x+30=-60x+80$$

Add 60x to both sides.

$$-4x^{2}-30x+30+60x=80$$

Combine -30x and 60x to get 30x.

$$-4x^{2}+30x+30=80$$

Subtract 80 from both sides.

$$-4x^{2}+30x+30-80=0$$

Subtract 80 from 30 to get -50.

$$-4x^{2}+30x-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 30 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-4\right)\left(-50\right)}}{2\left(-4\right)}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-4\right)\left(-50\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-30±\sqrt{900+16\left(-50\right)}}{2\left(-4\right)}$$

Multiply 16 times -50.

$$x=\frac{-30±\sqrt{900-800}}{2\left(-4\right)}$$

Add 900 to -800.

$$x=\frac{-30±\sqrt{100}}{2\left(-4\right)}$$

Take the square root of 100.

$$x=\frac{-30±10}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-30±10}{-8}$$

Now solve the equation x=\frac{-30±10}{-8} when ± is plus. Add -30 to 10.

$$x=-\frac{20}{-8}$$

Reduce the fraction \frac{-20}{-8} to lowest terms by extracting and canceling out 4.

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

Now solve the equation x=\frac{-30±10}{-8} when ± is minus. Subtract 10 from -30.

$$x=-\frac{40}{-8}$$

Divide -40 by -8.

$$x=5$$

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values 2,4 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-4\right)\left(x-2\right), the least common multiple of x-2,x-4,3.

$$\left(3x-12\right)\left(x-1\right)+\left(3x-6\right)\left(x-3\right)=\left(x-4\right)\left(x-2\right)\left(3\times 3+1\right)$$

Use the distributive property to multiply 3x-12 by x-1 and combine like terms.

$$3x^{2}-15x+12+\left(3x-6\right)\left(x-3\right)=\left(x-4\right)\left(x-2\right)\left(3\times 3+1\right)$$

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

$$3x^{2}-15x+12+3x^{2}-15x+18=\left(x-4\right)\left(x-2\right)\left(3\times 3+1\right)$$

Combine 3x^{2} and 3x^{2} to get 6x^{2}.

$$6x^{2}-15x+12-15x+18=\left(x-4\right)\left(x-2\right)\left(3\times 3+1\right)$$

Combine -15x and -15x to get -30x.

$$6x^{2}-30x+12+18=\left(x-4\right)\left(x-2\right)\left(3\times 3+1\right)$$

Add 12 and 18 to get 30.

$$6x^{2}-30x+30=\left(x-4\right)\left(x-2\right)\left(3\times 3+1\right)$$

Multiply 3 and 3 to get 9.

$$6x^{2}-30x+30=\left(x-4\right)\left(x-2\right)\left(9+1\right)$$

Add 9 and 1 to get 10.

$$6x^{2}-30x+30=\left(x-4\right)\left(x-2\right)\times 10$$

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

$$6x^{2}-30x+30=\left(x^{2}-6x+8\right)\times 10$$

Use the distributive property to multiply x^{2}-6x+8 by 10.

$$6x^{2}-30x+30=10x^{2}-60x+80$$

Subtract 10x^{2} from both sides.

$$6x^{2}-30x+30-10x^{2}=-60x+80$$

Combine 6x^{2} and -10x^{2} to get -4x^{2}.

$$-4x^{2}-30x+30=-60x+80$$

Add 60x to both sides.

$$-4x^{2}-30x+30+60x=80$$

Combine -30x and 60x to get 30x.

$$-4x^{2}+30x+30=80$$

Subtract 30 from both sides.

$$-4x^{2}+30x=80-30$$

Subtract 30 from 80 to get 50.

$$-4x^{2}+30x=50$$

Divide both sides by -4.

$$\frac{-4x^{2}+30x}{-4}=\frac{50}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{30}{-4}x=\frac{50}{-4}$$

Reduce the fraction \frac{30}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{15}{2}x=\frac{50}{-4}$$

Reduce the fraction \frac{50}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{15}{2}x=-\frac{25}{2}$$

Divide -\frac{15}{2}, the coefficient of the x term, by 2 to get -\frac{15}{4}. Then add the square of -\frac{15}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{2}x+\left(-\frac{15}{4}\right)^{2}=-\frac{25}{2}+\left(-\frac{15}{4}\right)^{2}$$

Square -\frac{15}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{2}x+\frac{225}{16}=-\frac{25}{2}+\frac{225}{16}$$

Add -\frac{25}{2} to \frac{225}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{2}x+\frac{225}{16}=\frac{25}{16}$$

Factor x^{2}-\frac{15}{2}x+\frac{225}{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}.

$$\left(x-\frac{15}{4}\right)^{2}=\frac{25}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Simplify.

$$x-\frac{15}{4}=\frac{5}{4} x-\frac{15}{4}=-\frac{5}{4}$$

Add \frac{15}{4} to both sides of the equation.

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