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

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

Solve for x (complex solution)

$$x=-\frac{3}{2}+\frac{1}{2}i=-1.5+0.5i$$

$$x=-\frac{3}{2}-\frac{1}{2}i=-1.5-0.5i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

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

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

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

Use the distributive property to multiply x+2 by 4.

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

Combine 4x and 4x to get 8x.

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

Add 8 and 1 to get 9.

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

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

$$2x^{2}+8x+9=2x+4$$

Subtract 2x from both sides.

$$2x^{2}+8x+9-2x=4$$

Combine 8x and -2x to get 6x.

$$2x^{2}+6x+9=4$$

Subtract 4 from both sides.

$$2x^{2}+6x+9-4=0$$

Subtract 4 from 9 to get 5.

$$2x^{2}+6x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 6 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 2\times 5}}{2\times 2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 2\times 5}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-6±\sqrt{36-8\times 5}}{2\times 2}$$

Multiply -8 times 5.

$$x=\frac{-6±\sqrt{36-40}}{2\times 2}$$

Add 36 to -40.

$$x=\frac{-6±\sqrt{-4}}{2\times 2}$$

Take the square root of -4.

$$x=\frac{-6±2i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-6±2i}{4}$$

Now solve the equation x=\frac{-6±2i}{4} when ± is plus. Add -6 to 2i.

$$x=\frac{-6+2i}{4}$$

Divide -6+2i by 4.

$$x=-\frac{3}{2}+\frac{1}{2}i$$

Now solve the equation x=\frac{-6±2i}{4} when ± is minus. Subtract 2i from -6.

$$x=\frac{-6-2i}{4}$$

Divide -6-2i by 4.

$$x=-\frac{3}{2}-\frac{1}{2}i$$

The equation is now solved.

$$x=-\frac{3}{2}+\frac{1}{2}i x=-\frac{3}{2}-\frac{1}{2}i$$

Steps for Completing the Square

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

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

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

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

Use the distributive property to multiply x+2 by 4.

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

Combine 4x and 4x to get 8x.

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

Add 8 and 1 to get 9.

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

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

$$2x^{2}+8x+9=2x+4$$

Subtract 2x from both sides.

$$2x^{2}+8x+9-2x=4$$

Combine 8x and -2x to get 6x.

$$2x^{2}+6x+9=4$$

Subtract 9 from both sides.

$$2x^{2}+6x=4-9$$

Subtract 9 from 4 to get -5.

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

Divide both sides by 2.

$$\frac{2x^{2}+6x}{2}=-\frac{5}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{6}{2}x=-\frac{5}{2}$$

Divide 6 by 2.

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

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=-\frac{5}{2}+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=-\frac{5}{2}+\frac{9}{4}$$

Add -\frac{5}{2} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=-\frac{1}{4}$$

Factor x^{2}+3x+\frac{9}{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}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{-\frac{1}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{1}{2}i x+\frac{3}{2}=-\frac{1}{2}i$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=-\frac{3}{2}+\frac{1}{2}i x=-\frac{3}{2}-\frac{1}{2}i$$