Solve 3/x-2=3/x-1+7/(x+1)(x+2)

$$\frac { 3 } { x - 2 } = \frac { 3 } { x - 1 } + \frac { 7 } { ( x + 1 ) ( x + 2 ) }$$

Solve for x

$$x = \frac{\sqrt{193} + 15}{4} \approx 7.223110997$$

$$x=\frac{15-\sqrt{193}}{4}\approx 0.276889003$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,-1,1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right), the least common multiple of x-2,x-1,\left(x+1\right)\left(x+2\right).

$$\left(x+2\right)\left(x^{2}-1\right)\times 3=\left(x+1\right)\left(x^{2}-4\right)\times 3+\left(x-2\right)\left(x-1\right)\times 7$$

Use the distributive property to multiply x+2 by x^{2}-1.

$$\left(x^{3}-x+2x^{2}-2\right)\times 3=\left(x+1\right)\left(x^{2}-4\right)\times 3+\left(x-2\right)\left(x-1\right)\times 7$$

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

$$3x^{3}-3x+6x^{2}-6=\left(x+1\right)\left(x^{2}-4\right)\times 3+\left(x-2\right)\left(x-1\right)\times 7$$

Use the distributive property to multiply x+1 by x^{2}-4.

$$3x^{3}-3x+6x^{2}-6=\left(x^{3}-4x+x^{2}-4\right)\times 3+\left(x-2\right)\left(x-1\right)\times 7$$

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

$$3x^{3}-3x+6x^{2}-6=3x^{3}-12x+3x^{2}-12+\left(x-2\right)\left(x-1\right)\times 7$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$3x^{3}-3x+6x^{2}-6=3x^{3}-12x+3x^{2}-12+\left(x^{2}-3x+2\right)\times 7$$

Use the distributive property to multiply x^{2}-3x+2 by 7.

$$3x^{3}-3x+6x^{2}-6=3x^{3}-12x+3x^{2}-12+7x^{2}-21x+14$$

Combine 3x^{2} and 7x^{2} to get 10x^{2}.

$$3x^{3}-3x+6x^{2}-6=3x^{3}-12x+10x^{2}-12-21x+14$$

Combine -12x and -21x to get -33x.

$$3x^{3}-3x+6x^{2}-6=3x^{3}-33x+10x^{2}-12+14$$

Add -12 and 14 to get 2.

$$3x^{3}-3x+6x^{2}-6=3x^{3}-33x+10x^{2}+2$$

Subtract 3x^{3} from both sides.

$$3x^{3}-3x+6x^{2}-6-3x^{3}=-33x+10x^{2}+2$$

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

$$-3x+6x^{2}-6=-33x+10x^{2}+2$$

Add 33x to both sides.

$$-3x+6x^{2}-6+33x=10x^{2}+2$$

Combine -3x and 33x to get 30x.

$$30x+6x^{2}-6=10x^{2}+2$$

Subtract 10x^{2} from both sides.

$$30x+6x^{2}-6-10x^{2}=2$$

Combine 6x^{2} and -10x^{2} to get -4x^{2}.

$$30x-4x^{2}-6=2$$

Subtract 2 from both sides.

$$30x-4x^{2}-6-2=0$$

Subtract 2 from -6 to get -8.

$$30x-4x^{2}-8=0$$

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.

$$-4x^{2}+30x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 30 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-4\right)\left(-8\right)}}{2\left(-4\right)}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-4\right)\left(-8\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-30±\sqrt{900+16\left(-8\right)}}{2\left(-4\right)}$$

Multiply 16 times -8.

$$x=\frac{-30±\sqrt{900-128}}{2\left(-4\right)}$$

Add 900 to -128.

$$x=\frac{-30±\sqrt{772}}{2\left(-4\right)}$$

Take the square root of 772.

$$x=\frac{-30±2\sqrt{193}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-30±2\sqrt{193}}{-8}$$

Now solve the equation x=\frac{-30±2\sqrt{193}}{-8} when ± is plus. Add -30 to 2\sqrt{193}.

$$x=\frac{2\sqrt{193}-30}{-8}$$

Divide -30+2\sqrt{193} by -8.

$$x=\frac{15-\sqrt{193}}{4}$$

Now solve the equation x=\frac{-30±2\sqrt{193}}{-8} when ± is minus. Subtract 2\sqrt{193} from -30.

$$x=\frac{-2\sqrt{193}-30}{-8}$$

Divide -30-2\sqrt{193} by -8.

$$x=\frac{\sqrt{193}+15}{4}$$

The equation is now solved.

$$x=\frac{15-\sqrt{193}}{4} x=\frac{\sqrt{193}+15}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,-1,1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right), the least common multiple of x-2,x-1,\left(x+1\right)\left(x+2\right).

$$\left(x+2\right)\left(x^{2}-1\right)\times 3=\left(x+1\right)\left(x^{2}-4\right)\times 3+\left(x-2\right)\left(x-1\right)\times 7$$

Use the distributive property to multiply x+2 by x^{2}-1.

$$\left(x^{3}-x+2x^{2}-2\right)\times 3=\left(x+1\right)\left(x^{2}-4\right)\times 3+\left(x-2\right)\left(x-1\right)\times 7$$

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

$$3x^{3}-3x+6x^{2}-6=\left(x+1\right)\left(x^{2}-4\right)\times 3+\left(x-2\right)\left(x-1\right)\times 7$$

Use the distributive property to multiply x+1 by x^{2}-4.

$$3x^{3}-3x+6x^{2}-6=\left(x^{3}-4x+x^{2}-4\right)\times 3+\left(x-2\right)\left(x-1\right)\times 7$$

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

$$3x^{3}-3x+6x^{2}-6=3x^{3}-12x+3x^{2}-12+\left(x-2\right)\left(x-1\right)\times 7$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$3x^{3}-3x+6x^{2}-6=3x^{3}-12x+3x^{2}-12+\left(x^{2}-3x+2\right)\times 7$$

Use the distributive property to multiply x^{2}-3x+2 by 7.

$$3x^{3}-3x+6x^{2}-6=3x^{3}-12x+3x^{2}-12+7x^{2}-21x+14$$

Combine 3x^{2} and 7x^{2} to get 10x^{2}.

$$3x^{3}-3x+6x^{2}-6=3x^{3}-12x+10x^{2}-12-21x+14$$

Combine -12x and -21x to get -33x.

$$3x^{3}-3x+6x^{2}-6=3x^{3}-33x+10x^{2}-12+14$$

Add -12 and 14 to get 2.

$$3x^{3}-3x+6x^{2}-6=3x^{3}-33x+10x^{2}+2$$

Subtract 3x^{3} from both sides.

$$3x^{3}-3x+6x^{2}-6-3x^{3}=-33x+10x^{2}+2$$

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

$$-3x+6x^{2}-6=-33x+10x^{2}+2$$

Add 33x to both sides.

$$-3x+6x^{2}-6+33x=10x^{2}+2$$

Combine -3x and 33x to get 30x.

$$30x+6x^{2}-6=10x^{2}+2$$

Subtract 10x^{2} from both sides.

$$30x+6x^{2}-6-10x^{2}=2$$

Combine 6x^{2} and -10x^{2} to get -4x^{2}.

$$30x-4x^{2}-6=2$$

Add 6 to both sides.

$$30x-4x^{2}=2+6$$

Add 2 and 6 to get 8.

$$30x-4x^{2}=8$$

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.

$$-4x^{2}+30x=8$$

Divide both sides by -4.

$$\frac{-4x^{2}+30x}{-4}=\frac{8}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{30}{-4}x=\frac{8}{-4}$$

Reduce the fraction \frac{30}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{15}{2}x=\frac{8}{-4}$$

Divide 8 by -4.

$$x^{2}-\frac{15}{2}x=-2$$

Divide -\frac{15}{2}, the coefficient of the x term, by 2 to get -\frac{15}{4}. Then add the square of -\frac{15}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{2}x+\left(-\frac{15}{4}\right)^{2}=-2+\left(-\frac{15}{4}\right)^{2}$$

Square -\frac{15}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{2}x+\frac{225}{16}=-2+\frac{225}{16}$$

Add -2 to \frac{225}{16}.

$$x^{2}-\frac{15}{2}x+\frac{225}{16}=\frac{193}{16}$$

Factor x^{2}-\frac{15}{2}x+\frac{225}{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}.

$$\left(x-\frac{15}{4}\right)^{2}=\frac{193}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{4}\right)^{2}}=\sqrt{\frac{193}{16}}$$

Simplify.

$$x-\frac{15}{4}=\frac{\sqrt{193}}{4} x-\frac{15}{4}=-\frac{\sqrt{193}}{4}$$

Add \frac{15}{4} to both sides of the equation.

$$x=\frac{\sqrt{193}+15}{4} x=\frac{15-\sqrt{193}}{4}$$