Solve x+1/x+2-x/x-1=a/x^2+x-2

$$\frac { x + 1 } { x + 2 } - \frac { x } { x - 1 } = \frac { a } { x ^ { 2 } + x - 2 }$$

Solve for x (complex solution)

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

$$a\neq 3\text{ and }a\neq -3$$

Steps for Solving Linear Equation

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

$$\left(x-1\right)\left(x+1\right)-\left(x+2\right)x=a$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$x^{2}-1-\left(x+2\right)x=a$$

Use the distributive property to multiply x+2 by x.

$$x^{2}-1-\left(x^{2}+2x\right)=a$$

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

$$x^{2}-1-x^{2}-2x=a$$

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

$$-1-2x=a$$

Add 1 to both sides.

$$-2x=a+1$$

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

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

Divide a+1 by -2.

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

Variable x cannot be equal to any of the values -2,1.

$$x=\frac{-a-1}{2}\text{, }x\neq -2\text{ and }x\neq 1$$

Solve for a

$$a=-2x-1$$

$$x\neq -2\text{ and }x\neq 1$$

Solution Steps

Multiply both sides of the equation by \left(x-1\right)\left(x+2\right), the least common multiple of x+2,x-1,x^{2}+x-2.

$$\left(x-1\right)\left(x+1\right)-\left(x+2\right)x=a$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$x^{2}-1-\left(x+2\right)x=a$$

Use the distributive property to multiply x+2 by x.

$$x^{2}-1-\left(x^{2}+2x\right)=a$$

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

$$x^{2}-1-x^{2}-2x=a$$

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

$$-1-2x=a$$

Swap sides so that all variable terms are on the left hand side.

$$a=-1-2x$$

Solve for x

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

$$|a|\neq 3$$

Steps for Solving Linear Equation

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

$$\left(x-1\right)\left(x+1\right)-\left(x+2\right)x=a$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$x^{2}-1-\left(x+2\right)x=a$$

Use the distributive property to multiply x+2 by x.

$$x^{2}-1-\left(x^{2}+2x\right)=a$$

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

$$x^{2}-1-x^{2}-2x=a$$

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

$$-1-2x=a$$

Add 1 to both sides.

$$-2x=a+1$$

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

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

Divide a+1 by -2.

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

Variable x cannot be equal to any of the values -2,1.

$$x=\frac{-a-1}{2}\text{, }x\neq -2\text{ and }x\neq 1$$