Solve 5/x-3-x-1/x-2=7

$$\frac { 5 } { x - 3 } - \frac { x - 1 } { x - 2 } = 7$$

Solve for x

$$x = \frac{\sqrt{11} + 11}{4} \approx 3.579156198$$

$$x = \frac{11 - \sqrt{11}}{4} \approx 1.920843802$$

Steps Using the Quadratic Formula

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

$$\left(x-2\right)\times 5-\left(x-3\right)\left(x-1\right)=7\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x-2 by 5.

$$5x-10-\left(x-3\right)\left(x-1\right)=7\left(x-3\right)\left(x-2\right)$$

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

$$5x-10-\left(x^{2}-4x+3\right)=7\left(x-3\right)\left(x-2\right)$$

To find the opposite of x^{2}-4x+3, find the opposite of each term.

$$5x-10-x^{2}+4x-3=7\left(x-3\right)\left(x-2\right)$$

Combine 5x and 4x to get 9x.

$$9x-10-x^{2}-3=7\left(x-3\right)\left(x-2\right)$$

Subtract 3 from -10 to get -13.

$$9x-13-x^{2}=7\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply 7 by x-3.

$$9x-13-x^{2}=\left(7x-21\right)\left(x-2\right)$$

Use the distributive property to multiply 7x-21 by x-2 and combine like terms.

$$9x-13-x^{2}=7x^{2}-35x+42$$

Subtract 7x^{2} from both sides.

$$9x-13-x^{2}-7x^{2}=-35x+42$$

Combine -x^{2} and -7x^{2} to get -8x^{2}.

$$9x-13-8x^{2}=-35x+42$$

Add 35x to both sides.

$$9x-13-8x^{2}+35x=42$$

Combine 9x and 35x to get 44x.

$$44x-13-8x^{2}=42$$

Subtract 42 from both sides.

$$44x-13-8x^{2}-42=0$$

Subtract 42 from -13 to get -55.

$$44x-55-8x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-8x^{2}+44x-55=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 44 for b, and -55 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-44±\sqrt{44^{2}-4\left(-8\right)\left(-55\right)}}{2\left(-8\right)}$$

Square 44.

$$x=\frac{-44±\sqrt{1936-4\left(-8\right)\left(-55\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-44±\sqrt{1936+32\left(-55\right)}}{2\left(-8\right)}$$

Multiply 32 times -55.

$$x=\frac{-44±\sqrt{1936-1760}}{2\left(-8\right)}$$

Add 1936 to -1760.

$$x=\frac{-44±\sqrt{176}}{2\left(-8\right)}$$

Take the square root of 176.

$$x=\frac{-44±4\sqrt{11}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-44±4\sqrt{11}}{-16}$$

Now solve the equation x=\frac{-44±4\sqrt{11}}{-16} when ± is plus. Add -44 to 4\sqrt{11}.

$$x=\frac{4\sqrt{11}-44}{-16}$$

Divide -44+4\sqrt{11} by -16.

$$x=\frac{11-\sqrt{11}}{4}$$

Now solve the equation x=\frac{-44±4\sqrt{11}}{-16} when ± is minus. Subtract 4\sqrt{11} from -44.

$$x=\frac{-4\sqrt{11}-44}{-16}$$

Divide -44-4\sqrt{11} by -16.

$$x=\frac{\sqrt{11}+11}{4}$$

The equation is now solved.

$$x=\frac{11-\sqrt{11}}{4} x=\frac{\sqrt{11}+11}{4}$$

Steps for Completing the Square

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

$$\left(x-2\right)\times 5-\left(x-3\right)\left(x-1\right)=7\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x-2 by 5.

$$5x-10-\left(x-3\right)\left(x-1\right)=7\left(x-3\right)\left(x-2\right)$$

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

$$5x-10-\left(x^{2}-4x+3\right)=7\left(x-3\right)\left(x-2\right)$$

To find the opposite of x^{2}-4x+3, find the opposite of each term.

$$5x-10-x^{2}+4x-3=7\left(x-3\right)\left(x-2\right)$$

Combine 5x and 4x to get 9x.

$$9x-10-x^{2}-3=7\left(x-3\right)\left(x-2\right)$$

Subtract 3 from -10 to get -13.

$$9x-13-x^{2}=7\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply 7 by x-3.

$$9x-13-x^{2}=\left(7x-21\right)\left(x-2\right)$$

Use the distributive property to multiply 7x-21 by x-2 and combine like terms.

$$9x-13-x^{2}=7x^{2}-35x+42$$

Subtract 7x^{2} from both sides.

$$9x-13-x^{2}-7x^{2}=-35x+42$$

Combine -x^{2} and -7x^{2} to get -8x^{2}.

$$9x-13-8x^{2}=-35x+42$$

Add 35x to both sides.

$$9x-13-8x^{2}+35x=42$$

Combine 9x and 35x to get 44x.

$$44x-13-8x^{2}=42$$

Add 13 to both sides.

$$44x-8x^{2}=42+13$$

Add 42 and 13 to get 55.

$$44x-8x^{2}=55$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-8x^{2}+44x=55$$

Divide both sides by -8.

$$\frac{-8x^{2}+44x}{-8}=\frac{55}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{44}{-8}x=\frac{55}{-8}$$

Reduce the fraction \frac{44}{-8} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{11}{2}x=\frac{55}{-8}$$

Divide 55 by -8.

$$x^{2}-\frac{11}{2}x=-\frac{55}{8}$$

Divide -\frac{11}{2}, the coefficient of the x term, by 2 to get -\frac{11}{4}. Then add the square of -\frac{11}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{2}x+\left(-\frac{11}{4}\right)^{2}=-\frac{55}{8}+\left(-\frac{11}{4}\right)^{2}$$

Square -\frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=-\frac{55}{8}+\frac{121}{16}$$

Add -\frac{55}{8} to \frac{121}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{11}{16}$$

Factor x^{2}-\frac{11}{2}x+\frac{121}{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}.

$$\left(x-\frac{11}{4}\right)^{2}=\frac{11}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{4}\right)^{2}}=\sqrt{\frac{11}{16}}$$

Simplify.

$$x-\frac{11}{4}=\frac{\sqrt{11}}{4} x-\frac{11}{4}=-\frac{\sqrt{11}}{4}$$

Add \frac{11}{4} to both sides of the equation.

$$x=\frac{\sqrt{11}+11}{4} x=\frac{11-\sqrt{11}}{4}$$