Solve x+2/x+1+x+3/x-2=29/(x+1)(x-2)

$$\frac { x + 2 } { x + 1 } + \frac { x + 3 } { x - 2 } = \frac { 29 } { ( x + 1 ) ( x - 2 ) }$$

Solve for x

$$x=-5$$

$$x=3$$

Steps Using the Quadratic Formula

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

$$\left(x-2\right)\left(x+2\right)+\left(x+1\right)\left(x+3\right)=29$$

Consider \left(x-2\right)\left(x+2\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 2.

$$x^{2}-4+\left(x+1\right)\left(x+3\right)=29$$

Use the distributive property to multiply x+1 by x+3 and combine like terms.

$$x^{2}-4+x^{2}+4x+3=29$$

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

$$2x^{2}-4+4x+3=29$$

Add -4 and 3 to get -1.

$$2x^{2}-1+4x=29$$

Subtract 29 from both sides.

$$2x^{2}-1+4x-29=0$$

Subtract 29 from -1 to get -30.

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

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.

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

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

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

Square 4.

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

Multiply -4 times 2.

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

Multiply -8 times -30.

$$x=\frac{-4±\sqrt{16+240}}{2\times 2}$$

Add 16 to 240.

$$x=\frac{-4±\sqrt{256}}{2\times 2}$$

Take the square root of 256.

$$x=\frac{-4±16}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±16}{4}$$

Now solve the equation x=\frac{-4±16}{4} when ± is plus. Add -4 to 16.

$$x=\frac{12}{4}$$

Divide 12 by 4.

$$x=3$$

Now solve the equation x=\frac{-4±16}{4} when ± is minus. Subtract 16 from -4.

$$x=-\frac{20}{4}$$

Divide -20 by 4.

$$x=-5$$

The equation is now solved.

$$x=3 x=-5$$

Steps for Completing the Square

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

$$\left(x-2\right)\left(x+2\right)+\left(x+1\right)\left(x+3\right)=29$$

Consider \left(x-2\right)\left(x+2\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 2.

$$x^{2}-4+\left(x+1\right)\left(x+3\right)=29$$

Use the distributive property to multiply x+1 by x+3 and combine like terms.

$$x^{2}-4+x^{2}+4x+3=29$$

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

$$2x^{2}-4+4x+3=29$$

Add -4 and 3 to get -1.

$$2x^{2}-1+4x=29$$

Add 1 to both sides.

$$2x^{2}+4x=29+1$$

Add 29 and 1 to get 30.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide 4 by 2.

$$x^{2}+2x=\frac{30}{2}$$

Divide 30 by 2.

$$x^{2}+2x=15$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=15+1^{2}$$

Square 1.

$$x^{2}+2x+1=15+1$$

Add 15 to 1.

$$x^{2}+2x+1=16$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+1=4 x+1=-4$$

Subtract 1 from both sides of the equation.

$$x=3 x=-5$$