Solve 1/x+1+1/x+2=13/12

$$\frac { 1 } { x + 1 } + \frac { 1 } { x + 2 } = \frac { 13 } { 12 }$$

Solve for x

$$x=\frac{\sqrt{745}-15}{26}\approx 0.47287262$$

$$x=\frac{-\sqrt{745}-15}{26}\approx -1.626718774$$

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 12\left(x+1\right)\left(x+2\right), the least common multiple of x+1,x+2,12.

$$12x+24+12x+12=13\left(x+1\right)\left(x+2\right)$$

Combine 12x and 12x to get 24x.

$$24x+24+12=13\left(x+1\right)\left(x+2\right)$$

Add 24 and 12 to get 36.

$$24x+36=13\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply 13 by x+1.

$$24x+36=\left(13x+13\right)\left(x+2\right)$$

Use the distributive property to multiply 13x+13 by x+2 and combine like terms.

$$24x+36=13x^{2}+39x+26$$

Subtract 13x^{2} from both sides.

$$24x+36-13x^{2}=39x+26$$

Subtract 39x from both sides.

$$24x+36-13x^{2}-39x=26$$

Combine 24x and -39x to get -15x.

$$-15x+36-13x^{2}=26$$

Subtract 26 from both sides.

$$-15x+36-13x^{2}-26=0$$

Subtract 26 from 36 to get 10.

$$-15x+10-13x^{2}=0$$

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.

$$-13x^{2}-15x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -13 for a, -15 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-13\right)\times 10}}{2\left(-13\right)}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-13\right)\times 10}}{2\left(-13\right)}$$

Multiply -4 times -13.

$$x=\frac{-\left(-15\right)±\sqrt{225+52\times 10}}{2\left(-13\right)}$$

Multiply 52 times 10.

$$x=\frac{-\left(-15\right)±\sqrt{225+520}}{2\left(-13\right)}$$

Add 225 to 520.

$$x=\frac{-\left(-15\right)±\sqrt{745}}{2\left(-13\right)}$$

The opposite of -15 is 15.

$$x=\frac{15±\sqrt{745}}{2\left(-13\right)}$$

Multiply 2 times -13.

$$x=\frac{15±\sqrt{745}}{-26}$$

Now solve the equation x=\frac{15±\sqrt{745}}{-26} when ± is plus. Add 15 to \sqrt{745}.

$$x=\frac{\sqrt{745}+15}{-26}$$

Divide 15+\sqrt{745} by -26.

$$x=\frac{-\sqrt{745}-15}{26}$$

Now solve the equation x=\frac{15±\sqrt{745}}{-26} when ± is minus. Subtract \sqrt{745} from 15.

$$x=\frac{15-\sqrt{745}}{-26}$$

Divide 15-\sqrt{745} by -26.

$$x=\frac{\sqrt{745}-15}{26}$$

The equation is now solved.

$$x=\frac{-\sqrt{745}-15}{26} x=\frac{\sqrt{745}-15}{26}$$

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 12\left(x+1\right)\left(x+2\right), the least common multiple of x+1,x+2,12.

$$12x+24+12x+12=13\left(x+1\right)\left(x+2\right)$$

Combine 12x and 12x to get 24x.

$$24x+24+12=13\left(x+1\right)\left(x+2\right)$$

Add 24 and 12 to get 36.

$$24x+36=13\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply 13 by x+1.

$$24x+36=\left(13x+13\right)\left(x+2\right)$$

Use the distributive property to multiply 13x+13 by x+2 and combine like terms.

$$24x+36=13x^{2}+39x+26$$

Subtract 13x^{2} from both sides.

$$24x+36-13x^{2}=39x+26$$

Subtract 39x from both sides.

$$24x+36-13x^{2}-39x=26$$

Combine 24x and -39x to get -15x.

$$-15x+36-13x^{2}=26$$

Subtract 36 from both sides.

$$-15x-13x^{2}=26-36$$

Subtract 36 from 26 to get -10.

$$-15x-13x^{2}=-10$$

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.

$$-13x^{2}-15x=-10$$

Divide both sides by -13.

$$\frac{-13x^{2}-15x}{-13}=-\frac{10}{-13}$$

Dividing by -13 undoes the multiplication by -13.

$$x^{2}+\left(-\frac{15}{-13}\right)x=-\frac{10}{-13}$$

Divide -15 by -13.

$$x^{2}+\frac{15}{13}x=-\frac{10}{-13}$$

Divide -10 by -13.

$$x^{2}+\frac{15}{13}x=\frac{10}{13}$$

Divide \frac{15}{13}, the coefficient of the x term, by 2 to get \frac{15}{26}. Then add the square of \frac{15}{26} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{15}{13}x+\left(\frac{15}{26}\right)^{2}=\frac{10}{13}+\left(\frac{15}{26}\right)^{2}$$

Square \frac{15}{26} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{15}{13}x+\frac{225}{676}=\frac{10}{13}+\frac{225}{676}$$

Add \frac{10}{13} to \frac{225}{676} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{15}{13}x+\frac{225}{676}=\frac{745}{676}$$

Factor x^{2}+\frac{15}{13}x+\frac{225}{676}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{15}{26}\right)^{2}=\frac{745}{676}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{26}\right)^{2}}=\sqrt{\frac{745}{676}}$$

Simplify.

$$x+\frac{15}{26}=\frac{\sqrt{745}}{26} x+\frac{15}{26}=-\frac{\sqrt{745}}{26}$$

Subtract \frac{15}{26} from both sides of the equation.

$$x=\frac{\sqrt{745}-15}{26} x=\frac{-\sqrt{745}-15}{26}$$