Solve 1/x+1+2/x+2=9/2

$$\frac { 1 } { x + 1 } + \frac { 2 } { x + 2 } = \frac { 9 } { 2 }$$

Solve for x

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

$$x=-\frac{2}{3}\approx -0.666666667$$

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

$$2x+4+\left(2x+2\right)\times 2=9\left(x+1\right)\left(x+2\right)$$

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

$$2x+4+4x+4=9\left(x+1\right)\left(x+2\right)$$

Combine 2x and 4x to get 6x.

$$6x+4+4=9\left(x+1\right)\left(x+2\right)$$

Add 4 and 4 to get 8.

$$6x+8=9\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply 9 by x+1.

$$6x+8=\left(9x+9\right)\left(x+2\right)$$

Use the distributive property to multiply 9x+9 by x+2 and combine like terms.

$$6x+8=9x^{2}+27x+18$$

Subtract 9x^{2} from both sides.

$$6x+8-9x^{2}=27x+18$$

Subtract 27x from both sides.

$$6x+8-9x^{2}-27x=18$$

Combine 6x and -27x to get -21x.

$$-21x+8-9x^{2}=18$$

Subtract 18 from both sides.

$$-21x+8-9x^{2}-18=0$$

Subtract 18 from 8 to get -10.

$$-21x-10-9x^{2}=0$$

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.

$$-9x^{2}-21x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -21 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\left(-9\right)\left(-10\right)}}{2\left(-9\right)}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\left(-9\right)\left(-10\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-21\right)±\sqrt{441+36\left(-10\right)}}{2\left(-9\right)}$$

Multiply 36 times -10.

$$x=\frac{-\left(-21\right)±\sqrt{441-360}}{2\left(-9\right)}$$

Add 441 to -360.

$$x=\frac{-\left(-21\right)±\sqrt{81}}{2\left(-9\right)}$$

Take the square root of 81.

$$x=\frac{-\left(-21\right)±9}{2\left(-9\right)}$$

The opposite of -21 is 21.

$$x=\frac{21±9}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{21±9}{-18}$$

Now solve the equation x=\frac{21±9}{-18} when ± is plus. Add 21 to 9.

$$x=\frac{30}{-18}$$

Reduce the fraction \frac{30}{-18} to lowest terms by extracting and canceling out 6.

$$x=-\frac{5}{3}$$

Now solve the equation x=\frac{21±9}{-18} when ± is minus. Subtract 9 from 21.

$$x=\frac{12}{-18}$$

Reduce the fraction \frac{12}{-18} to lowest terms by extracting and canceling out 6.

$$x=-\frac{2}{3}$$

The equation is now solved.

$$x=-\frac{5}{3} x=-\frac{2}{3}$$

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

$$2x+4+\left(2x+2\right)\times 2=9\left(x+1\right)\left(x+2\right)$$

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

$$2x+4+4x+4=9\left(x+1\right)\left(x+2\right)$$

Combine 2x and 4x to get 6x.

$$6x+4+4=9\left(x+1\right)\left(x+2\right)$$

Add 4 and 4 to get 8.

$$6x+8=9\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply 9 by x+1.

$$6x+8=\left(9x+9\right)\left(x+2\right)$$

Use the distributive property to multiply 9x+9 by x+2 and combine like terms.

$$6x+8=9x^{2}+27x+18$$

Subtract 9x^{2} from both sides.

$$6x+8-9x^{2}=27x+18$$

Subtract 27x from both sides.

$$6x+8-9x^{2}-27x=18$$

Combine 6x and -27x to get -21x.

$$-21x+8-9x^{2}=18$$

Subtract 8 from both sides.

$$-21x-9x^{2}=18-8$$

Subtract 8 from 18 to get 10.

$$-21x-9x^{2}=10$$

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.

$$-9x^{2}-21x=10$$

Divide both sides by -9.

$$\frac{-9x^{2}-21x}{-9}=\frac{10}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\left(-\frac{21}{-9}\right)x=\frac{10}{-9}$$

Reduce the fraction \frac{-21}{-9} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{7}{3}x=\frac{10}{-9}$$

Divide 10 by -9.

$$x^{2}+\frac{7}{3}x=-\frac{10}{9}$$

Divide \frac{7}{3}, the coefficient of the x term, by 2 to get \frac{7}{6}. Then add the square of \frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=-\frac{10}{9}+\left(\frac{7}{6}\right)^{2}$$

Square \frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=-\frac{10}{9}+\frac{49}{36}$$

Add -\frac{10}{9} to \frac{49}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{1}{4}$$

Factor x^{2}+\frac{7}{3}x+\frac{49}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

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

Subtract \frac{7}{6} from both sides of the equation.

$$x=-\frac{2}{3} x=-\frac{5}{3}$$