Solve 1/x-4-1/x+4=1/x-5-1/x-2

$$\frac { 1 } { x - 4 } - \frac { 1 } { x + 4 } = \frac { 1 } { x - 5 } - \frac { 1 } { x - 2 }$$

Solve for x

$$x=8$$

$$x = \frac{16}{5} = 3\frac{1}{5} = 3.2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,2,4,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x-4\right)\left(x-2\right)\left(x+4\right), the least common multiple of x-4,x+4,x-5,x-2.

$$\left(x-5\right)\left(x-2\right)\left(x+4\right)-\left(x-5\right)\left(x-4\right)\left(x-2\right)=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Use the distributive property to multiply x-5 by x-2 and combine like terms.

$$\left(x^{2}-7x+10\right)\left(x+4\right)-\left(x-5\right)\left(x-4\right)\left(x-2\right)=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Use the distributive property to multiply x^{2}-7x+10 by x+4 and combine like terms.

$$x^{3}-3x^{2}-18x+40-\left(x-5\right)\left(x-4\right)\left(x-2\right)=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Use the distributive property to multiply x-5 by x-4 and combine like terms.

$$x^{3}-3x^{2}-18x+40-\left(x^{2}-9x+20\right)\left(x-2\right)=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Use the distributive property to multiply x^{2}-9x+20 by x-2 and combine like terms.

$$x^{3}-3x^{2}-18x+40-\left(x^{3}-11x^{2}+38x-40\right)=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

To find the opposite of x^{3}-11x^{2}+38x-40, find the opposite of each term.

$$x^{3}-3x^{2}-18x+40-x^{3}+11x^{2}-38x+40=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

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

$$-3x^{2}-18x+40+11x^{2}-38x+40=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

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

$$8x^{2}-18x+40-38x+40=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Combine -18x and -38x to get -56x.

$$8x^{2}-56x+40+40=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Add 40 and 40 to get 80.

$$8x^{2}-56x+80=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Use the distributive property to multiply x-2 by x^{2}-16.

$$8x^{2}-56x+80=x^{3}-16x-2x^{2}+32-\left(x-5\right)\left(x^{2}-16\right)$$

Use the distributive property to multiply x-5 by x^{2}-16.

$$8x^{2}-56x+80=x^{3}-16x-2x^{2}+32-\left(x^{3}-16x-5x^{2}+80\right)$$

To find the opposite of x^{3}-16x-5x^{2}+80, find the opposite of each term.

$$8x^{2}-56x+80=x^{3}-16x-2x^{2}+32-x^{3}+16x+5x^{2}-80$$

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

$$8x^{2}-56x+80=-16x-2x^{2}+32+16x+5x^{2}-80$$

Combine -16x and 16x to get 0.

$$8x^{2}-56x+80=-2x^{2}+32+5x^{2}-80$$

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

$$8x^{2}-56x+80=3x^{2}+32-80$$

Subtract 80 from 32 to get -48.

$$8x^{2}-56x+80=3x^{2}-48$$

Subtract 3x^{2} from both sides.

$$8x^{2}-56x+80-3x^{2}=-48$$

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

$$5x^{2}-56x+80=-48$$

Add 48 to both sides.

$$5x^{2}-56x+80+48=0$$

Add 80 and 48 to get 128.

$$5x^{2}-56x+128=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -56 for b, and 128 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\times 5\times 128}}{2\times 5}$$

Square -56.

$$x=\frac{-\left(-56\right)±\sqrt{3136-4\times 5\times 128}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-56\right)±\sqrt{3136-20\times 128}}{2\times 5}$$

Multiply -20 times 128.

$$x=\frac{-\left(-56\right)±\sqrt{3136-2560}}{2\times 5}$$

Add 3136 to -2560.

$$x=\frac{-\left(-56\right)±\sqrt{576}}{2\times 5}$$

Take the square root of 576.

$$x=\frac{-\left(-56\right)±24}{2\times 5}$$

The opposite of -56 is 56.

$$x=\frac{56±24}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{56±24}{10}$$

Now solve the equation x=\frac{56±24}{10} when ± is plus. Add 56 to 24.

$$x=\frac{80}{10}$$

Divide 80 by 10.

$$x=8$$

Now solve the equation x=\frac{56±24}{10} when ± is minus. Subtract 24 from 56.

$$x=\frac{32}{10}$$

Reduce the fraction \frac{32}{10} to lowest terms by extracting and canceling out 2.

$$x=\frac{16}{5}$$

The equation is now solved.

$$x=8 x=\frac{16}{5}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,2,4,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x-4\right)\left(x-2\right)\left(x+4\right), the least common multiple of x-4,x+4,x-5,x-2.

$$\left(x-5\right)\left(x-2\right)\left(x+4\right)-\left(x-5\right)\left(x-4\right)\left(x-2\right)=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Use the distributive property to multiply x-5 by x-2 and combine like terms.

$$\left(x^{2}-7x+10\right)\left(x+4\right)-\left(x-5\right)\left(x-4\right)\left(x-2\right)=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Use the distributive property to multiply x^{2}-7x+10 by x+4 and combine like terms.

$$x^{3}-3x^{2}-18x+40-\left(x-5\right)\left(x-4\right)\left(x-2\right)=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Use the distributive property to multiply x-5 by x-4 and combine like terms.

$$x^{3}-3x^{2}-18x+40-\left(x^{2}-9x+20\right)\left(x-2\right)=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Use the distributive property to multiply x^{2}-9x+20 by x-2 and combine like terms.

$$x^{3}-3x^{2}-18x+40-\left(x^{3}-11x^{2}+38x-40\right)=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

To find the opposite of x^{3}-11x^{2}+38x-40, find the opposite of each term.

$$x^{3}-3x^{2}-18x+40-x^{3}+11x^{2}-38x+40=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

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

$$-3x^{2}-18x+40+11x^{2}-38x+40=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

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

$$8x^{2}-18x+40-38x+40=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Combine -18x and -38x to get -56x.

$$8x^{2}-56x+40+40=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Add 40 and 40 to get 80.

$$8x^{2}-56x+80=\left(x-2\right)\left(x^{2}-16\right)-\left(x-5\right)\left(x^{2}-16\right)$$

Use the distributive property to multiply x-2 by x^{2}-16.

$$8x^{2}-56x+80=x^{3}-16x-2x^{2}+32-\left(x-5\right)\left(x^{2}-16\right)$$

Use the distributive property to multiply x-5 by x^{2}-16.

$$8x^{2}-56x+80=x^{3}-16x-2x^{2}+32-\left(x^{3}-16x-5x^{2}+80\right)$$

To find the opposite of x^{3}-16x-5x^{2}+80, find the opposite of each term.

$$8x^{2}-56x+80=x^{3}-16x-2x^{2}+32-x^{3}+16x+5x^{2}-80$$

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

$$8x^{2}-56x+80=-16x-2x^{2}+32+16x+5x^{2}-80$$

Combine -16x and 16x to get 0.

$$8x^{2}-56x+80=-2x^{2}+32+5x^{2}-80$$

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

$$8x^{2}-56x+80=3x^{2}+32-80$$

Subtract 80 from 32 to get -48.

$$8x^{2}-56x+80=3x^{2}-48$$

Subtract 3x^{2} from both sides.

$$8x^{2}-56x+80-3x^{2}=-48$$

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

$$5x^{2}-56x+80=-48$$

Subtract 80 from both sides.

$$5x^{2}-56x=-48-80$$

Subtract 80 from -48 to get -128.

$$5x^{2}-56x=-128$$

Divide both sides by 5.

$$\frac{5x^{2}-56x}{5}=-\frac{128}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{56}{5}x=-\frac{128}{5}$$

Divide -\frac{56}{5}, the coefficient of the x term, by 2 to get -\frac{28}{5}. Then add the square of -\frac{28}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{56}{5}x+\left(-\frac{28}{5}\right)^{2}=-\frac{128}{5}+\left(-\frac{28}{5}\right)^{2}$$

Square -\frac{28}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{56}{5}x+\frac{784}{25}=-\frac{128}{5}+\frac{784}{25}$$

Add -\frac{128}{5} to \frac{784}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{56}{5}x+\frac{784}{25}=\frac{144}{25}$$

Factor x^{2}-\frac{56}{5}x+\frac{784}{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}.

$$\left(x-\frac{28}{5}\right)^{2}=\frac{144}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{28}{5}\right)^{2}}=\sqrt{\frac{144}{25}}$$

Simplify.

$$x-\frac{28}{5}=\frac{12}{5} x-\frac{28}{5}=-\frac{12}{5}$$

Add \frac{28}{5} to both sides of the equation.

$$x=8 x=\frac{16}{5}$$