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

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

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{47}i}{2}\approx -0.5+3.4278273i$$

$$x=\frac{-\sqrt{47}i-1}{2}\approx -0.5-3.4278273i$$

Steps Using the Quadratic Formula

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

$$\left(2x+8\right)\times 2+\left(2x+2\right)x=\left(x+1\right)\left(x+4\right)$$

Use the distributive property to multiply 2x+8 by 2.

$$4x+16+\left(2x+2\right)x=\left(x+1\right)\left(x+4\right)$$

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

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

Combine 4x and 2x to get 6x.

$$6x+16+2x^{2}=\left(x+1\right)\left(x+4\right)$$

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

$$6x+16+2x^{2}=x^{2}+5x+4$$

Subtract x^{2} from both sides.

$$6x+16+2x^{2}-x^{2}=5x+4$$

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

$$6x+16+x^{2}=5x+4$$

Subtract 5x from both sides.

$$6x+16+x^{2}-5x=4$$

Combine 6x and -5x to get x.

$$x+16+x^{2}=4$$

Subtract 4 from both sides.

$$x+16+x^{2}-4=0$$

Subtract 4 from 16 to get 12.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 12}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 12}}{2}$$

Multiply -4 times 12.

$$x=\frac{-1±\sqrt{1-48}}{2}$$

Add 1 to -48.

$$x=\frac{-1±\sqrt{-47}}{2}$$

Take the square root of -47.

$$x=\frac{-1±\sqrt{47}i}{2}$$

Now solve the equation x=\frac{-1±\sqrt{47}i}{2} when ± is plus. Add -1 to i\sqrt{47}.

$$x=\frac{-1+\sqrt{47}i}{2}$$

Now solve the equation x=\frac{-1±\sqrt{47}i}{2} when ± is minus. Subtract i\sqrt{47} from -1.

$$x=\frac{-\sqrt{47}i-1}{2}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{47}i}{2} x=\frac{-\sqrt{47}i-1}{2}$$

Steps for Completing the Square

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

$$\left(2x+8\right)\times 2+\left(2x+2\right)x=\left(x+1\right)\left(x+4\right)$$

Use the distributive property to multiply 2x+8 by 2.

$$4x+16+\left(2x+2\right)x=\left(x+1\right)\left(x+4\right)$$

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

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

Combine 4x and 2x to get 6x.

$$6x+16+2x^{2}=\left(x+1\right)\left(x+4\right)$$

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

$$6x+16+2x^{2}=x^{2}+5x+4$$

Subtract x^{2} from both sides.

$$6x+16+2x^{2}-x^{2}=5x+4$$

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

$$6x+16+x^{2}=5x+4$$

Subtract 5x from both sides.

$$6x+16+x^{2}-5x=4$$

Combine 6x and -5x to get x.

$$x+16+x^{2}=4$$

Subtract 16 from both sides.

$$x+x^{2}=4-16$$

Subtract 16 from 4 to get -12.

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

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.

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

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=-12+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=-12+\frac{1}{4}$$

Add -12 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=-\frac{47}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=-\frac{47}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{-\frac{47}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{47}i}{2} x+\frac{1}{2}=-\frac{\sqrt{47}i}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{-1+\sqrt{47}i}{2} x=\frac{-\sqrt{47}i-1}{2}$$