Solve x+1/x+2+x+2/x+1=7/6

$$\frac { x + 1 } { x + 2 } + \frac { x + 2 } { x + 1 } = \frac { 7 } { 6 }$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{95}i}{10}-\frac{3}{2}\approx -1.5-0.974679434i$$

$$x=\frac{\sqrt{95}i}{10}-\frac{3}{2}\approx -1.5+0.974679434i$$

Steps Using the Quadratic Formula

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 6\left(x+1\right)\left(x+2\right), the least common multiple of x+2,x+1,6.

$$\left(6x+6\right)\left(x+1\right)+\left(6x+12\right)\left(x+2\right)=7\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply 6x+6 by x+1 and combine like terms.

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

Use the distributive property to multiply 6x+12 by x+2 and combine like terms.

$$6x^{2}+12x+6+6x^{2}+24x+24=7\left(x+1\right)\left(x+2\right)$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$12x^{2}+12x+6+24x+24=7\left(x+1\right)\left(x+2\right)$$

Combine 12x and 24x to get 36x.

$$12x^{2}+36x+6+24=7\left(x+1\right)\left(x+2\right)$$

Add 6 and 24 to get 30.

$$12x^{2}+36x+30=7\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply 7 by x+1.

$$12x^{2}+36x+30=\left(7x+7\right)\left(x+2\right)$$

Use the distributive property to multiply 7x+7 by x+2 and combine like terms.

$$12x^{2}+36x+30=7x^{2}+21x+14$$

Subtract 7x^{2} from both sides.

$$12x^{2}+36x+30-7x^{2}=21x+14$$

Combine 12x^{2} and -7x^{2} to get 5x^{2}.

$$5x^{2}+36x+30=21x+14$$

Subtract 21x from both sides.

$$5x^{2}+36x+30-21x=14$$

Combine 36x and -21x to get 15x.

$$5x^{2}+15x+30=14$$

Subtract 14 from both sides.

$$5x^{2}+15x+30-14=0$$

Subtract 14 from 30 to get 16.

$$5x^{2}+15x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 15 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 5\times 16}}{2\times 5}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 5\times 16}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-15±\sqrt{225-20\times 16}}{2\times 5}$$

Multiply -20 times 16.

$$x=\frac{-15±\sqrt{225-320}}{2\times 5}$$

Add 225 to -320.

$$x=\frac{-15±\sqrt{-95}}{2\times 5}$$

Take the square root of -95.

$$x=\frac{-15±\sqrt{95}i}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-15±\sqrt{95}i}{10}$$

Now solve the equation x=\frac{-15±\sqrt{95}i}{10} when ± is plus. Add -15 to i\sqrt{95}.

$$x=\frac{-15+\sqrt{95}i}{10}$$

Divide -15+i\sqrt{95} by 10.

$$x=\frac{\sqrt{95}i}{10}-\frac{3}{2}$$

Now solve the equation x=\frac{-15±\sqrt{95}i}{10} when ± is minus. Subtract i\sqrt{95} from -15.

$$x=\frac{-\sqrt{95}i-15}{10}$$

Divide -15-i\sqrt{95} by 10.

$$x=-\frac{\sqrt{95}i}{10}-\frac{3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{95}i}{10}-\frac{3}{2} x=-\frac{\sqrt{95}i}{10}-\frac{3}{2}$$

Steps for Completing the Square

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 6\left(x+1\right)\left(x+2\right), the least common multiple of x+2,x+1,6.

$$\left(6x+6\right)\left(x+1\right)+\left(6x+12\right)\left(x+2\right)=7\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply 6x+6 by x+1 and combine like terms.

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

Use the distributive property to multiply 6x+12 by x+2 and combine like terms.

$$6x^{2}+12x+6+6x^{2}+24x+24=7\left(x+1\right)\left(x+2\right)$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$12x^{2}+12x+6+24x+24=7\left(x+1\right)\left(x+2\right)$$

Combine 12x and 24x to get 36x.

$$12x^{2}+36x+6+24=7\left(x+1\right)\left(x+2\right)$$

Add 6 and 24 to get 30.

$$12x^{2}+36x+30=7\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply 7 by x+1.

$$12x^{2}+36x+30=\left(7x+7\right)\left(x+2\right)$$

Use the distributive property to multiply 7x+7 by x+2 and combine like terms.

$$12x^{2}+36x+30=7x^{2}+21x+14$$

Subtract 7x^{2} from both sides.

$$12x^{2}+36x+30-7x^{2}=21x+14$$

Combine 12x^{2} and -7x^{2} to get 5x^{2}.

$$5x^{2}+36x+30=21x+14$$

Subtract 21x from both sides.

$$5x^{2}+36x+30-21x=14$$

Combine 36x and -21x to get 15x.

$$5x^{2}+15x+30=14$$

Subtract 30 from both sides.

$$5x^{2}+15x=14-30$$

Subtract 30 from 14 to get -16.

$$5x^{2}+15x=-16$$

Divide both sides by 5.

$$\frac{5x^{2}+15x}{5}=-\frac{16}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{15}{5}x=-\frac{16}{5}$$

Divide 15 by 5.

$$x^{2}+3x=-\frac{16}{5}$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=-\frac{16}{5}+\frac{9}{4}$$

Add -\frac{16}{5} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=-\frac{19}{20}$$

Factor x^{2}+3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{2}\right)^{2}=-\frac{19}{20}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{-\frac{19}{20}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{95}i}{10} x+\frac{3}{2}=-\frac{\sqrt{95}i}{10}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{\sqrt{95}i}{10}-\frac{3}{2} x=-\frac{\sqrt{95}i}{10}-\frac{3}{2}$$