Solve +1/x+1+1/x+3+1/x+4=0

$$+ \frac { 1 } { x + 1 } + \frac { 1 } { x + 3 } + \frac { 1 } { x + 4 } = 0$$

Solve for x

$$x=\frac{\sqrt{7}-8}{3}\approx -1.784749563$$

$$x=\frac{-\sqrt{7}-8}{3}\approx -3.54858377$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,-3,-1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)\left(x+3\right)\left(x+4\right), the least common multiple of x+1,x+3,x+4.

$$\left(x+3\right)\left(x+4\right)+\left(x+1\right)\left(x+4\right)+\left(x+1\right)\left(x+3\right)=0$$

Use the distributive property to multiply x+3 by x+4 and combine like terms.

$$x^{2}+7x+12+\left(x+1\right)\left(x+4\right)+\left(x+1\right)\left(x+3\right)=0$$

Use the distributive property to multiply x+1 by x+4 and combine like terms.

$$x^{2}+7x+12+x^{2}+5x+4+\left(x+1\right)\left(x+3\right)=0$$

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

$$2x^{2}+7x+12+5x+4+\left(x+1\right)\left(x+3\right)=0$$

Combine 7x and 5x to get 12x.

$$2x^{2}+12x+12+4+\left(x+1\right)\left(x+3\right)=0$$

Add 12 and 4 to get 16.

$$2x^{2}+12x+16+\left(x+1\right)\left(x+3\right)=0$$

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

$$2x^{2}+12x+16+x^{2}+4x+3=0$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$3x^{2}+12x+16+4x+3=0$$

Combine 12x and 4x to get 16x.

$$3x^{2}+16x+16+3=0$$

Add 16 and 3 to get 19.

$$3x^{2}+16x+19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 16 for b, and 19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 3\times 19}}{2\times 3}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 3\times 19}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-16±\sqrt{256-12\times 19}}{2\times 3}$$

Multiply -12 times 19.

$$x=\frac{-16±\sqrt{256-228}}{2\times 3}$$

Add 256 to -228.

$$x=\frac{-16±\sqrt{28}}{2\times 3}$$

Take the square root of 28.

$$x=\frac{-16±2\sqrt{7}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-16±2\sqrt{7}}{6}$$

Now solve the equation x=\frac{-16±2\sqrt{7}}{6} when ± is plus. Add -16 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}-16}{6}$$

Divide -16+2\sqrt{7} by 6.

$$x=\frac{\sqrt{7}-8}{3}$$

Now solve the equation x=\frac{-16±2\sqrt{7}}{6} when ± is minus. Subtract 2\sqrt{7} from -16.

$$x=\frac{-2\sqrt{7}-16}{6}$$

Divide -16-2\sqrt{7} by 6.

$$x=\frac{-\sqrt{7}-8}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{7}-8}{3} x=\frac{-\sqrt{7}-8}{3}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,-3,-1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)\left(x+3\right)\left(x+4\right), the least common multiple of x+1,x+3,x+4.

$$\left(x+3\right)\left(x+4\right)+\left(x+1\right)\left(x+4\right)+\left(x+1\right)\left(x+3\right)=0$$

Use the distributive property to multiply x+3 by x+4 and combine like terms.

$$x^{2}+7x+12+\left(x+1\right)\left(x+4\right)+\left(x+1\right)\left(x+3\right)=0$$

Use the distributive property to multiply x+1 by x+4 and combine like terms.

$$x^{2}+7x+12+x^{2}+5x+4+\left(x+1\right)\left(x+3\right)=0$$

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

$$2x^{2}+7x+12+5x+4+\left(x+1\right)\left(x+3\right)=0$$

Combine 7x and 5x to get 12x.

$$2x^{2}+12x+12+4+\left(x+1\right)\left(x+3\right)=0$$

Add 12 and 4 to get 16.

$$2x^{2}+12x+16+\left(x+1\right)\left(x+3\right)=0$$

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

$$2x^{2}+12x+16+x^{2}+4x+3=0$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$3x^{2}+12x+16+4x+3=0$$

Combine 12x and 4x to get 16x.

$$3x^{2}+16x+16+3=0$$

Add 16 and 3 to get 19.

$$3x^{2}+16x+19=0$$

Subtract 19 from both sides. Anything subtracted from zero gives its negation.

$$3x^{2}+16x=-19$$

Divide both sides by 3.

$$\frac{3x^{2}+16x}{3}=-\frac{19}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{16}{3}x=-\frac{19}{3}$$

Divide \frac{16}{3}, the coefficient of the x term, by 2 to get \frac{8}{3}. Then add the square of \frac{8}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{16}{3}x+\left(\frac{8}{3}\right)^{2}=-\frac{19}{3}+\left(\frac{8}{3}\right)^{2}$$

Square \frac{8}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=-\frac{19}{3}+\frac{64}{9}$$

Add -\frac{19}{3} to \frac{64}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=\frac{7}{9}$$

Factor x^{2}+\frac{16}{3}x+\frac{64}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8}{3}\right)^{2}=\frac{7}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8}{3}\right)^{2}}=\sqrt{\frac{7}{9}}$$

Simplify.

$$x+\frac{8}{3}=\frac{\sqrt{7}}{3} x+\frac{8}{3}=-\frac{\sqrt{7}}{3}$$

Subtract \frac{8}{3} from both sides of the equation.

$$x=\frac{\sqrt{7}-8}{3} x=\frac{-\sqrt{7}-8}{3}$$