Solve 1/x+1+1/x+2=4/x+4

$$\frac{ 1 }{ x+1 } + \frac{ 1 }{ x+2 } = \frac{ 4 }{ x+4 }$$

Solve for x

$$x = \frac{\sqrt{33} - 1}{4} \approx 1.186140662$$

$$x=\frac{-\sqrt{33}-1}{4}\approx -1.686140662$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,-2,-1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)\left(x+2\right)\left(x+4\right), the least common multiple of x+1,x+2,x+4.

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

Use the distributive property to multiply x+2 by x+4 and combine like terms.

$$x^{2}+6x+8+\left(x+1\right)\left(x+4\right)=\left(x+1\right)\left(x+2\right)\times 4$$

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

$$x^{2}+6x+8+x^{2}+5x+4=\left(x+1\right)\left(x+2\right)\times 4$$

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

$$2x^{2}+6x+8+5x+4=\left(x+1\right)\left(x+2\right)\times 4$$

Combine 6x and 5x to get 11x.

$$2x^{2}+11x+8+4=\left(x+1\right)\left(x+2\right)\times 4$$

Add 8 and 4 to get 12.

$$2x^{2}+11x+12=\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x+1 by x+2 and combine like terms.

$$2x^{2}+11x+12=\left(x^{2}+3x+2\right)\times 4$$

Use the distributive property to multiply x^{2}+3x+2 by 4.

$$2x^{2}+11x+12=4x^{2}+12x+8$$

Subtract 4x^{2} from both sides.

$$2x^{2}+11x+12-4x^{2}=12x+8$$

Combine 2x^{2} and -4x^{2} to get -2x^{2}.

$$-2x^{2}+11x+12=12x+8$$

Subtract 12x from both sides.

$$-2x^{2}+11x+12-12x=8$$

Combine 11x and -12x to get -x.

$$-2x^{2}-x+12=8$$

Subtract 8 from both sides.

$$-2x^{2}-x+12-8=0$$

Subtract 8 from 12 to get 4.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -1 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-2\right)\times 4}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-1\right)±\sqrt{1+8\times 4}}{2\left(-2\right)}$$

Multiply 8 times 4.

$$x=\frac{-\left(-1\right)±\sqrt{1+32}}{2\left(-2\right)}$$

Add 1 to 32.

$$x=\frac{-\left(-1\right)±\sqrt{33}}{2\left(-2\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{33}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{1±\sqrt{33}}{-4}$$

Now solve the equation x=\frac{1±\sqrt{33}}{-4} when ± is plus. Add 1 to \sqrt{33}.

$$x=\frac{\sqrt{33}+1}{-4}$$

Divide 1+\sqrt{33} by -4.

$$x=\frac{-\sqrt{33}-1}{4}$$

Now solve the equation x=\frac{1±\sqrt{33}}{-4} when ± is minus. Subtract \sqrt{33} from 1.

$$x=\frac{1-\sqrt{33}}{-4}$$

Divide 1-\sqrt{33} by -4.

$$x=\frac{\sqrt{33}-1}{4}$$

The equation is now solved.

$$x=\frac{-\sqrt{33}-1}{4} x=\frac{\sqrt{33}-1}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,-2,-1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)\left(x+2\right)\left(x+4\right), the least common multiple of x+1,x+2,x+4.

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

Use the distributive property to multiply x+2 by x+4 and combine like terms.

$$x^{2}+6x+8+\left(x+1\right)\left(x+4\right)=\left(x+1\right)\left(x+2\right)\times 4$$

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

$$x^{2}+6x+8+x^{2}+5x+4=\left(x+1\right)\left(x+2\right)\times 4$$

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

$$2x^{2}+6x+8+5x+4=\left(x+1\right)\left(x+2\right)\times 4$$

Combine 6x and 5x to get 11x.

$$2x^{2}+11x+8+4=\left(x+1\right)\left(x+2\right)\times 4$$

Add 8 and 4 to get 12.

$$2x^{2}+11x+12=\left(x+1\right)\left(x+2\right)\times 4$$

Use the distributive property to multiply x+1 by x+2 and combine like terms.

$$2x^{2}+11x+12=\left(x^{2}+3x+2\right)\times 4$$

Use the distributive property to multiply x^{2}+3x+2 by 4.

$$2x^{2}+11x+12=4x^{2}+12x+8$$

Subtract 4x^{2} from both sides.

$$2x^{2}+11x+12-4x^{2}=12x+8$$

Combine 2x^{2} and -4x^{2} to get -2x^{2}.

$$-2x^{2}+11x+12=12x+8$$

Subtract 12x from both sides.

$$-2x^{2}+11x+12-12x=8$$

Combine 11x and -12x to get -x.

$$-2x^{2}-x+12=8$$

Subtract 12 from both sides.

$$-2x^{2}-x=8-12$$

Subtract 12 from 8 to get -4.

$$-2x^{2}-x=-4$$

Divide both sides by -2.

$$\frac{-2x^{2}-x}{-2}=-\frac{4}{-2}$$

Dividing by -2 undoes the multiplication by -2.

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

Divide -1 by -2.

$$x^{2}+\frac{1}{2}x=-\frac{4}{-2}$$

Divide -4 by -2.

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

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

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

Add 2 to \frac{1}{16}.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{33}{16}$$

Factor x^{2}+\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{33}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{33}{16}}$$

Simplify.

$$x+\frac{1}{4}=\frac{\sqrt{33}}{4} x+\frac{1}{4}=-\frac{\sqrt{33}}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

$$x=\frac{\sqrt{33}-1}{4} x=\frac{-\sqrt{33}-1}{4}$$