Solve 1/x-6-1/x+2=1/x-7-1/x-4

$$\frac { 1 } { x - 6 } - \frac { 1 } { x + 2 } = \frac { 1 } { x - 7 } - \frac { 1 } { x - 4 }$$

Solve for x

$$x=10$$

$$x = \frac{26}{5} = 5\frac{1}{5} = 5.2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,4,6,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x-6\right)\left(x-4\right)\left(x+2\right), the least common multiple of x-6,x+2,x-7,x-4.

$$\left(x-7\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x-4\right)=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x-7 by x-4 and combine like terms.

$$\left(x^{2}-11x+28\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x-4\right)=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x^{2}-11x+28 by x+2 and combine like terms.

$$x^{3}-9x^{2}+6x+56-\left(x-7\right)\left(x-6\right)\left(x-4\right)=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x-7 by x-6 and combine like terms.

$$x^{3}-9x^{2}+6x+56-\left(x^{2}-13x+42\right)\left(x-4\right)=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x^{2}-13x+42 by x-4 and combine like terms.

$$x^{3}-9x^{2}+6x+56-\left(x^{3}-17x^{2}+94x-168\right)=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

To find the opposite of x^{3}-17x^{2}+94x-168, find the opposite of each term.

$$x^{3}-9x^{2}+6x+56-x^{3}+17x^{2}-94x+168=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Combine x^{3} and -x^{3} to get 0.

$$-9x^{2}+6x+56+17x^{2}-94x+168=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Combine -9x^{2} and 17x^{2} to get 8x^{2}.

$$8x^{2}+6x+56-94x+168=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Combine 6x and -94x to get -88x.

$$8x^{2}-88x+56+168=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Add 56 and 168 to get 224.

$$8x^{2}-88x+224=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x-6 by x-4 and combine like terms.

$$8x^{2}-88x+224=\left(x^{2}-10x+24\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x^{2}-10x+24 by x+2 and combine like terms.

$$8x^{2}-88x+224=x^{3}-8x^{2}+4x+48-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x-7 by x-6 and combine like terms.

$$8x^{2}-88x+224=x^{3}-8x^{2}+4x+48-\left(x^{2}-13x+42\right)\left(x+2\right)$$

Use the distributive property to multiply x^{2}-13x+42 by x+2 and combine like terms.

$$8x^{2}-88x+224=x^{3}-8x^{2}+4x+48-\left(x^{3}-11x^{2}+16x+84\right)$$

To find the opposite of x^{3}-11x^{2}+16x+84, find the opposite of each term.

$$8x^{2}-88x+224=x^{3}-8x^{2}+4x+48-x^{3}+11x^{2}-16x-84$$

Combine x^{3} and -x^{3} to get 0.

$$8x^{2}-88x+224=-8x^{2}+4x+48+11x^{2}-16x-84$$

Combine -8x^{2} and 11x^{2} to get 3x^{2}.

$$8x^{2}-88x+224=3x^{2}+4x+48-16x-84$$

Combine 4x and -16x to get -12x.

$$8x^{2}-88x+224=3x^{2}-12x+48-84$$

Subtract 84 from 48 to get -36.

$$8x^{2}-88x+224=3x^{2}-12x-36$$

Subtract 3x^{2} from both sides.

$$8x^{2}-88x+224-3x^{2}=-12x-36$$

Combine 8x^{2} and -3x^{2} to get 5x^{2}.

$$5x^{2}-88x+224=-12x-36$$

Add 12x to both sides.

$$5x^{2}-88x+224+12x=-36$$

Combine -88x and 12x to get -76x.

$$5x^{2}-76x+224=-36$$

Add 36 to both sides.

$$5x^{2}-76x+224+36=0$$

Add 224 and 36 to get 260.

$$5x^{2}-76x+260=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -76 for b, and 260 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-76\right)±\sqrt{\left(-76\right)^{2}-4\times 5\times 260}}{2\times 5}$$

Square -76.

$$x=\frac{-\left(-76\right)±\sqrt{5776-4\times 5\times 260}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-76\right)±\sqrt{5776-20\times 260}}{2\times 5}$$

Multiply -20 times 260.

$$x=\frac{-\left(-76\right)±\sqrt{5776-5200}}{2\times 5}$$

Add 5776 to -5200.

$$x=\frac{-\left(-76\right)±\sqrt{576}}{2\times 5}$$

Take the square root of 576.

$$x=\frac{-\left(-76\right)±24}{2\times 5}$$

The opposite of -76 is 76.

$$x=\frac{76±24}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{76±24}{10}$$

Now solve the equation x=\frac{76±24}{10} when ± is plus. Add 76 to 24.

$$x=\frac{100}{10}$$

Divide 100 by 10.

$$x=10$$

Now solve the equation x=\frac{76±24}{10} when ± is minus. Subtract 24 from 76.

$$x=\frac{52}{10}$$

Reduce the fraction \frac{52}{10} to lowest terms by extracting and canceling out 2.

$$x=\frac{26}{5}$$

The equation is now solved.

$$x=10 x=\frac{26}{5}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,4,6,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x-6\right)\left(x-4\right)\left(x+2\right), the least common multiple of x-6,x+2,x-7,x-4.

$$\left(x-7\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x-4\right)=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x-7 by x-4 and combine like terms.

$$\left(x^{2}-11x+28\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x-4\right)=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x^{2}-11x+28 by x+2 and combine like terms.

$$x^{3}-9x^{2}+6x+56-\left(x-7\right)\left(x-6\right)\left(x-4\right)=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x-7 by x-6 and combine like terms.

$$x^{3}-9x^{2}+6x+56-\left(x^{2}-13x+42\right)\left(x-4\right)=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x^{2}-13x+42 by x-4 and combine like terms.

$$x^{3}-9x^{2}+6x+56-\left(x^{3}-17x^{2}+94x-168\right)=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

To find the opposite of x^{3}-17x^{2}+94x-168, find the opposite of each term.

$$x^{3}-9x^{2}+6x+56-x^{3}+17x^{2}-94x+168=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Combine x^{3} and -x^{3} to get 0.

$$-9x^{2}+6x+56+17x^{2}-94x+168=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Combine -9x^{2} and 17x^{2} to get 8x^{2}.

$$8x^{2}+6x+56-94x+168=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Combine 6x and -94x to get -88x.

$$8x^{2}-88x+56+168=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Add 56 and 168 to get 224.

$$8x^{2}-88x+224=\left(x-6\right)\left(x-4\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x-6 by x-4 and combine like terms.

$$8x^{2}-88x+224=\left(x^{2}-10x+24\right)\left(x+2\right)-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x^{2}-10x+24 by x+2 and combine like terms.

$$8x^{2}-88x+224=x^{3}-8x^{2}+4x+48-\left(x-7\right)\left(x-6\right)\left(x+2\right)$$

Use the distributive property to multiply x-7 by x-6 and combine like terms.

$$8x^{2}-88x+224=x^{3}-8x^{2}+4x+48-\left(x^{2}-13x+42\right)\left(x+2\right)$$

Use the distributive property to multiply x^{2}-13x+42 by x+2 and combine like terms.

$$8x^{2}-88x+224=x^{3}-8x^{2}+4x+48-\left(x^{3}-11x^{2}+16x+84\right)$$

To find the opposite of x^{3}-11x^{2}+16x+84, find the opposite of each term.

$$8x^{2}-88x+224=x^{3}-8x^{2}+4x+48-x^{3}+11x^{2}-16x-84$$

Combine x^{3} and -x^{3} to get 0.

$$8x^{2}-88x+224=-8x^{2}+4x+48+11x^{2}-16x-84$$

Combine -8x^{2} and 11x^{2} to get 3x^{2}.

$$8x^{2}-88x+224=3x^{2}+4x+48-16x-84$$

Combine 4x and -16x to get -12x.

$$8x^{2}-88x+224=3x^{2}-12x+48-84$$

Subtract 84 from 48 to get -36.

$$8x^{2}-88x+224=3x^{2}-12x-36$$

Subtract 3x^{2} from both sides.

$$8x^{2}-88x+224-3x^{2}=-12x-36$$

Combine 8x^{2} and -3x^{2} to get 5x^{2}.

$$5x^{2}-88x+224=-12x-36$$

Add 12x to both sides.

$$5x^{2}-88x+224+12x=-36$$

Combine -88x and 12x to get -76x.

$$5x^{2}-76x+224=-36$$

Subtract 224 from both sides.

$$5x^{2}-76x=-36-224$$

Subtract 224 from -36 to get -260.

$$5x^{2}-76x=-260$$

Divide both sides by 5.

$$\frac{5x^{2}-76x}{5}=-\frac{260}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{76}{5}x=-\frac{260}{5}$$

Divide -260 by 5.

$$x^{2}-\frac{76}{5}x=-52$$

Divide -\frac{76}{5}, the coefficient of the x term, by 2 to get -\frac{38}{5}. Then add the square of -\frac{38}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{76}{5}x+\left(-\frac{38}{5}\right)^{2}=-52+\left(-\frac{38}{5}\right)^{2}$$

Square -\frac{38}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{76}{5}x+\frac{1444}{25}=-52+\frac{1444}{25}$$

Add -52 to \frac{1444}{25}.

$$x^{2}-\frac{76}{5}x+\frac{1444}{25}=\frac{144}{25}$$

Factor x^{2}-\frac{76}{5}x+\frac{1444}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{38}{5}\right)^{2}=\frac{144}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{38}{5}\right)^{2}}=\sqrt{\frac{144}{25}}$$

Simplify.

$$x-\frac{38}{5}=\frac{12}{5} x-\frac{38}{5}=-\frac{12}{5}$$

Add \frac{38}{5} to both sides of the equation.

$$x=10 x=\frac{26}{5}$$