Solve 2/(x-1)(x-3)=-5+1/x-3

$$\frac { 2 } { ( x - 1 ) ( x - 3 ) } = - 5 + \frac { 1 } { x - 3 }$$

Solve for x

$$x = \frac{6}{5} = 1\frac{1}{5} = 1.2$$

Steps Using the Quadratic Formula

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

$$2=\left(x-3\right)\left(x-1\right)\left(-5\right)+x-1$$

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

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

Use the distributive property to multiply x^{2}-4x+3 by -5.

$$2=-5x^{2}+20x-15+x-1$$

Combine 20x and x to get 21x.

$$2=-5x^{2}+21x-15-1$$

Subtract 1 from -15 to get -16.

$$2=-5x^{2}+21x-16$$

Swap sides so that all variable terms are on the left hand side.

$$-5x^{2}+21x-16=2$$

Subtract 2 from both sides.

$$-5x^{2}+21x-16-2=0$$

Subtract 2 from -16 to get -18.

$$-5x^{2}+21x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 21 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-21±\sqrt{21^{2}-4\left(-5\right)\left(-18\right)}}{2\left(-5\right)}$$

Square 21.

$$x=\frac{-21±\sqrt{441-4\left(-5\right)\left(-18\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-21±\sqrt{441+20\left(-18\right)}}{2\left(-5\right)}$$

Multiply 20 times -18.

$$x=\frac{-21±\sqrt{441-360}}{2\left(-5\right)}$$

Add 441 to -360.

$$x=\frac{-21±\sqrt{81}}{2\left(-5\right)}$$

Take the square root of 81.

$$x=\frac{-21±9}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-21±9}{-10}$$

Now solve the equation x=\frac{-21±9}{-10} when ± is plus. Add -21 to 9.

$$x=-\frac{12}{-10}$$

Reduce the fraction \frac{-12}{-10} to lowest terms by extracting and canceling out 2.

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

Now solve the equation x=\frac{-21±9}{-10} when ± is minus. Subtract 9 from -21.

$$x=-\frac{30}{-10}$$

Divide -30 by -10.

$$x=3$$

The equation is now solved.

$$x=\frac{6}{5} x=3$$

Variable x cannot be equal to 3.

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

Steps for Completing the Square

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

$$2=\left(x-3\right)\left(x-1\right)\left(-5\right)+x-1$$

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

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

Use the distributive property to multiply x^{2}-4x+3 by -5.

$$2=-5x^{2}+20x-15+x-1$$

Combine 20x and x to get 21x.

$$2=-5x^{2}+21x-15-1$$

Subtract 1 from -15 to get -16.

$$2=-5x^{2}+21x-16$$

Swap sides so that all variable terms are on the left hand side.

$$-5x^{2}+21x-16=2$$

Add 16 to both sides.

$$-5x^{2}+21x=2+16$$

Add 2 and 16 to get 18.

$$-5x^{2}+21x=18$$

Divide both sides by -5.

$$\frac{-5x^{2}+21x}{-5}=\frac{18}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{21}{-5}x=\frac{18}{-5}$$

Divide 21 by -5.

$$x^{2}-\frac{21}{5}x=\frac{18}{-5}$$

Divide 18 by -5.

$$x^{2}-\frac{21}{5}x=-\frac{18}{5}$$

Divide -\frac{21}{5}, the coefficient of the x term, by 2 to get -\frac{21}{10}. Then add the square of -\frac{21}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{21}{5}x+\left(-\frac{21}{10}\right)^{2}=-\frac{18}{5}+\left(-\frac{21}{10}\right)^{2}$$

Square -\frac{21}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{21}{5}x+\frac{441}{100}=-\frac{18}{5}+\frac{441}{100}$$

Add -\frac{18}{5} to \frac{441}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{21}{5}x+\frac{441}{100}=\frac{81}{100}$$

Factor x^{2}-\frac{21}{5}x+\frac{441}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{10}\right)^{2}=\frac{81}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{10}\right)^{2}}=\sqrt{\frac{81}{100}}$$

Simplify.

$$x-\frac{21}{10}=\frac{9}{10} x-\frac{21}{10}=-\frac{9}{10}$$

Add \frac{21}{10} to both sides of the equation.

$$x=3 x=\frac{6}{5}$$

Variable x cannot be equal to 3.

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