Solve x-1/x+2-x-2/x+3=x-4/x+5-x-5/x+6

$$\frac{ x-1 }{ x+2 } - \frac{ x-2 }{ x+3 } = \frac{ x-4 }{ x+5 } - \frac{ x-5 }{ x+6 }$$

Solve for x

$$x=-4$$

$$x=-\frac{1}{2}=-0.5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -6,-5,-3,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x+6\right), the least common multiple of x+2,x+3,x+5,x+6.

$$\left(x+3\right)\left(x+5\right)\left(x+6\right)\left(x-1\right)-\left(x+2\right)\left(x+5\right)\left(x+6\right)\left(x-2\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x+3 by x+5 and combine like terms.

$$\left(x^{2}+8x+15\right)\left(x+6\right)\left(x-1\right)-\left(x+2\right)\left(x+5\right)\left(x+6\right)\left(x-2\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{2}+8x+15 by x+6 and combine like terms.

$$\left(x^{3}+14x^{2}+63x+90\right)\left(x-1\right)-\left(x+2\right)\left(x+5\right)\left(x+6\right)\left(x-2\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{3}+14x^{2}+63x+90 by x-1 and combine like terms.

$$x^{4}+13x^{3}+49x^{2}+27x-90-\left(x+2\right)\left(x+5\right)\left(x+6\right)\left(x-2\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

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

$$x^{4}+13x^{3}+49x^{2}+27x-90-\left(x^{2}+7x+10\right)\left(x+6\right)\left(x-2\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{2}+7x+10 by x+6 and combine like terms.

$$x^{4}+13x^{3}+49x^{2}+27x-90-\left(x^{3}+13x^{2}+52x+60\right)\left(x-2\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{3}+13x^{2}+52x+60 by x-2 and combine like terms.

$$x^{4}+13x^{3}+49x^{2}+27x-90-\left(x^{4}+11x^{3}+26x^{2}-44x-120\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

To find the opposite of x^{4}+11x^{3}+26x^{2}-44x-120, find the opposite of each term.

$$x^{4}+13x^{3}+49x^{2}+27x-90-x^{4}-11x^{3}-26x^{2}+44x+120=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Combine x^{4} and -x^{4} to get 0.

$$13x^{3}+49x^{2}+27x-90-11x^{3}-26x^{2}+44x+120=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Combine 13x^{3} and -11x^{3} to get 2x^{3}.

$$2x^{3}+49x^{2}+27x-90-26x^{2}+44x+120=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Combine 49x^{2} and -26x^{2} to get 23x^{2}.

$$2x^{3}+23x^{2}+27x-90+44x+120=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Combine 27x and 44x to get 71x.

$$2x^{3}+23x^{2}+71x-90+120=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Add -90 and 120 to get 30.

$$2x^{3}+23x^{2}+71x+30=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$2x^{3}+23x^{2}+71x+30=\left(x^{2}+5x+6\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{2}+5x+6 by x+6 and combine like terms.

$$2x^{3}+23x^{2}+71x+30=\left(x^{3}+11x^{2}+36x+36\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{3}+11x^{2}+36x+36 by x-4 and combine like terms.

$$2x^{3}+23x^{2}+71x+30=x^{4}+7x^{3}-8x^{2}-108x-144-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$2x^{3}+23x^{2}+71x+30=x^{4}+7x^{3}-8x^{2}-108x-144-\left(x^{2}+5x+6\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{2}+5x+6 by x+5 and combine like terms.

$$2x^{3}+23x^{2}+71x+30=x^{4}+7x^{3}-8x^{2}-108x-144-\left(x^{3}+10x^{2}+31x+30\right)\left(x-5\right)$$

Use the distributive property to multiply x^{3}+10x^{2}+31x+30 by x-5 and combine like terms.

$$2x^{3}+23x^{2}+71x+30=x^{4}+7x^{3}-8x^{2}-108x-144-\left(x^{4}+5x^{3}-19x^{2}-125x-150\right)$$

To find the opposite of x^{4}+5x^{3}-19x^{2}-125x-150, find the opposite of each term.

$$2x^{3}+23x^{2}+71x+30=x^{4}+7x^{3}-8x^{2}-108x-144-x^{4}-5x^{3}+19x^{2}+125x+150$$

Combine x^{4} and -x^{4} to get 0.

$$2x^{3}+23x^{2}+71x+30=7x^{3}-8x^{2}-108x-144-5x^{3}+19x^{2}+125x+150$$

Combine 7x^{3} and -5x^{3} to get 2x^{3}.

$$2x^{3}+23x^{2}+71x+30=2x^{3}-8x^{2}-108x-144+19x^{2}+125x+150$$

Combine -8x^{2} and 19x^{2} to get 11x^{2}.

$$2x^{3}+23x^{2}+71x+30=2x^{3}+11x^{2}-108x-144+125x+150$$

Combine -108x and 125x to get 17x.

$$2x^{3}+23x^{2}+71x+30=2x^{3}+11x^{2}+17x-144+150$$

Add -144 and 150 to get 6.

$$2x^{3}+23x^{2}+71x+30=2x^{3}+11x^{2}+17x+6$$

Subtract 2x^{3} from both sides.

$$2x^{3}+23x^{2}+71x+30-2x^{3}=11x^{2}+17x+6$$

Combine 2x^{3} and -2x^{3} to get 0.

$$23x^{2}+71x+30=11x^{2}+17x+6$$

Subtract 11x^{2} from both sides.

$$23x^{2}+71x+30-11x^{2}=17x+6$$

Combine 23x^{2} and -11x^{2} to get 12x^{2}.

$$12x^{2}+71x+30=17x+6$$

Subtract 17x from both sides.

$$12x^{2}+71x+30-17x=6$$

Combine 71x and -17x to get 54x.

$$12x^{2}+54x+30=6$$

Subtract 6 from both sides.

$$12x^{2}+54x+30-6=0$$

Subtract 6 from 30 to get 24.

$$12x^{2}+54x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 54 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-54±\sqrt{54^{2}-4\times 12\times 24}}{2\times 12}$$

Square 54.

$$x=\frac{-54±\sqrt{2916-4\times 12\times 24}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-54±\sqrt{2916-48\times 24}}{2\times 12}$$

Multiply -48 times 24.

$$x=\frac{-54±\sqrt{2916-1152}}{2\times 12}$$

Add 2916 to -1152.

$$x=\frac{-54±\sqrt{1764}}{2\times 12}$$

Take the square root of 1764.

$$x=\frac{-54±42}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-54±42}{24}$$

Now solve the equation x=\frac{-54±42}{24} when ± is plus. Add -54 to 42.

$$x=-\frac{12}{24}$$

Reduce the fraction \frac{-12}{24} to lowest terms by extracting and canceling out 12.

$$x=-\frac{1}{2}$$

Now solve the equation x=\frac{-54±42}{24} when ± is minus. Subtract 42 from -54.

$$x=-\frac{96}{24}$$

Divide -96 by 24.

$$x=-4$$

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -6,-5,-3,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x+6\right), the least common multiple of x+2,x+3,x+5,x+6.

$$\left(x+3\right)\left(x+5\right)\left(x+6\right)\left(x-1\right)-\left(x+2\right)\left(x+5\right)\left(x+6\right)\left(x-2\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x+3 by x+5 and combine like terms.

$$\left(x^{2}+8x+15\right)\left(x+6\right)\left(x-1\right)-\left(x+2\right)\left(x+5\right)\left(x+6\right)\left(x-2\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{2}+8x+15 by x+6 and combine like terms.

$$\left(x^{3}+14x^{2}+63x+90\right)\left(x-1\right)-\left(x+2\right)\left(x+5\right)\left(x+6\right)\left(x-2\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{3}+14x^{2}+63x+90 by x-1 and combine like terms.

$$x^{4}+13x^{3}+49x^{2}+27x-90-\left(x+2\right)\left(x+5\right)\left(x+6\right)\left(x-2\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

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

$$x^{4}+13x^{3}+49x^{2}+27x-90-\left(x^{2}+7x+10\right)\left(x+6\right)\left(x-2\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{2}+7x+10 by x+6 and combine like terms.

$$x^{4}+13x^{3}+49x^{2}+27x-90-\left(x^{3}+13x^{2}+52x+60\right)\left(x-2\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{3}+13x^{2}+52x+60 by x-2 and combine like terms.

$$x^{4}+13x^{3}+49x^{2}+27x-90-\left(x^{4}+11x^{3}+26x^{2}-44x-120\right)=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

To find the opposite of x^{4}+11x^{3}+26x^{2}-44x-120, find the opposite of each term.

$$x^{4}+13x^{3}+49x^{2}+27x-90-x^{4}-11x^{3}-26x^{2}+44x+120=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Combine x^{4} and -x^{4} to get 0.

$$13x^{3}+49x^{2}+27x-90-11x^{3}-26x^{2}+44x+120=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Combine 13x^{3} and -11x^{3} to get 2x^{3}.

$$2x^{3}+49x^{2}+27x-90-26x^{2}+44x+120=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Combine 49x^{2} and -26x^{2} to get 23x^{2}.

$$2x^{3}+23x^{2}+27x-90+44x+120=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Combine 27x and 44x to get 71x.

$$2x^{3}+23x^{2}+71x-90+120=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Add -90 and 120 to get 30.

$$2x^{3}+23x^{2}+71x+30=\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$2x^{3}+23x^{2}+71x+30=\left(x^{2}+5x+6\right)\left(x+6\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{2}+5x+6 by x+6 and combine like terms.

$$2x^{3}+23x^{2}+71x+30=\left(x^{3}+11x^{2}+36x+36\right)\left(x-4\right)-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{3}+11x^{2}+36x+36 by x-4 and combine like terms.

$$2x^{3}+23x^{2}+71x+30=x^{4}+7x^{3}-8x^{2}-108x-144-\left(x+2\right)\left(x+3\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$2x^{3}+23x^{2}+71x+30=x^{4}+7x^{3}-8x^{2}-108x-144-\left(x^{2}+5x+6\right)\left(x+5\right)\left(x-5\right)$$

Use the distributive property to multiply x^{2}+5x+6 by x+5 and combine like terms.

$$2x^{3}+23x^{2}+71x+30=x^{4}+7x^{3}-8x^{2}-108x-144-\left(x^{3}+10x^{2}+31x+30\right)\left(x-5\right)$$

Use the distributive property to multiply x^{3}+10x^{2}+31x+30 by x-5 and combine like terms.

$$2x^{3}+23x^{2}+71x+30=x^{4}+7x^{3}-8x^{2}-108x-144-\left(x^{4}+5x^{3}-19x^{2}-125x-150\right)$$

To find the opposite of x^{4}+5x^{3}-19x^{2}-125x-150, find the opposite of each term.

$$2x^{3}+23x^{2}+71x+30=x^{4}+7x^{3}-8x^{2}-108x-144-x^{4}-5x^{3}+19x^{2}+125x+150$$

Combine x^{4} and -x^{4} to get 0.

$$2x^{3}+23x^{2}+71x+30=7x^{3}-8x^{2}-108x-144-5x^{3}+19x^{2}+125x+150$$

Combine 7x^{3} and -5x^{3} to get 2x^{3}.

$$2x^{3}+23x^{2}+71x+30=2x^{3}-8x^{2}-108x-144+19x^{2}+125x+150$$

Combine -8x^{2} and 19x^{2} to get 11x^{2}.

$$2x^{3}+23x^{2}+71x+30=2x^{3}+11x^{2}-108x-144+125x+150$$

Combine -108x and 125x to get 17x.

$$2x^{3}+23x^{2}+71x+30=2x^{3}+11x^{2}+17x-144+150$$

Add -144 and 150 to get 6.

$$2x^{3}+23x^{2}+71x+30=2x^{3}+11x^{2}+17x+6$$

Subtract 2x^{3} from both sides.

$$2x^{3}+23x^{2}+71x+30-2x^{3}=11x^{2}+17x+6$$

Combine 2x^{3} and -2x^{3} to get 0.

$$23x^{2}+71x+30=11x^{2}+17x+6$$

Subtract 11x^{2} from both sides.

$$23x^{2}+71x+30-11x^{2}=17x+6$$

Combine 23x^{2} and -11x^{2} to get 12x^{2}.

$$12x^{2}+71x+30=17x+6$$

Subtract 17x from both sides.

$$12x^{2}+71x+30-17x=6$$

Combine 71x and -17x to get 54x.

$$12x^{2}+54x+30=6$$

Subtract 30 from both sides.

$$12x^{2}+54x=6-30$$

Subtract 30 from 6 to get -24.

$$12x^{2}+54x=-24$$

Divide both sides by 12.

$$\frac{12x^{2}+54x}{12}=-\frac{24}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\frac{54}{12}x=-\frac{24}{12}$$

Reduce the fraction \frac{54}{12} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{9}{2}x=-\frac{24}{12}$$

Divide -24 by 12.

$$x^{2}+\frac{9}{2}x=-2$$

Divide \frac{9}{2}, the coefficient of the x term, by 2 to get \frac{9}{4}. Then add the square of \frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=-2+\frac{81}{16}$$

Add -2 to \frac{81}{16}.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=\frac{49}{16}$$

Factor x^{2}+\frac{9}{2}x+\frac{81}{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}.

$$\left(x+\frac{9}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x+\frac{9}{4}=\frac{7}{4} x+\frac{9}{4}=-\frac{7}{4}$$

Subtract \frac{9}{4} from both sides of the equation.

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