Solve 1/2x+3/2x=1+1/2x

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

Solve for x (complex solution)

$$x=1+i$$

$$x=1-i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x, the least common multiple of 2,2x.

$$\frac{1}{2}x\times 2x+3=2x+1$$

Multiply \frac{1}{2} and 2 to get 1.

$$xx+3=2x+1$$

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

$$x^{2}+3=2x+1$$

Subtract 2x from both sides.

$$x^{2}+3-2x=1$$

Subtract 1 from both sides.

$$x^{2}+3-2x-1=0$$

Subtract 1 from 3 to get 2.

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

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.

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

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

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

Square -2.

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

Multiply -4 times 2.

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

Add 4 to -8.

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

Take the square root of -4.

$$x=\frac{-\left(-2\right)±2i}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±2i}{2}$$

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

$$x=\frac{2+2i}{2}$$

Divide 2+2i by 2.

$$x=1+i$$

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

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

Divide 2-2i by 2.

$$x=1-i$$

The equation is now solved.

$$x=1+i x=1-i$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x, the least common multiple of 2,2x.

$$\frac{1}{2}x\times 2x+3=2x+1$$

Multiply \frac{1}{2} and 2 to get 1.

$$xx+3=2x+1$$

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

$$x^{2}+3=2x+1$$

Subtract 2x from both sides.

$$x^{2}+3-2x=1$$

Subtract 3 from both sides.

$$x^{2}-2x=1-3$$

Subtract 3 from 1 to get -2.

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

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

$$x^{2}-2x+1=-2+1$$

Add -2 to 1.

$$x^{2}-2x+1=-1$$

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

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-1}$$

Simplify.

$$x-1=i x-1=-i$$

Add 1 to both sides of the equation.

$$x=1+i x=1-i$$