Solve 1/2x=3-x/2

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

Solve for x

$$x=2\sqrt{2}+3\approx 5.828427125$$

$$x=3-2\sqrt{2}\approx 0.171572875$$

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 2x,2.

$$1=2x\times 3-xx$$

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

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

Multiply 2 and 3 to get 6.

$$1=6x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$6x-x^{2}=1$$

Subtract 1 from both sides.

$$6x-x^{2}-1=0$$

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.

$$-x^{2}+6x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 6 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-6±\sqrt{36+4\left(-1\right)}}{2\left(-1\right)}$$

Multiply 4 times -1.

$$x=\frac{-6±\sqrt{36-4}}{2\left(-1\right)}$$

Add 36 to -4.

$$x=\frac{-6±\sqrt{32}}{2\left(-1\right)}$$

Take the square root of 32.

$$x=\frac{-6±4\sqrt{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-6±4\sqrt{2}}{-2}$$

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

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

Divide -6+4\sqrt{2} by -2.

$$x=3-2\sqrt{2}$$

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

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

Divide -6-4\sqrt{2} by -2.

$$x=2\sqrt{2}+3$$

The equation is now solved.

$$x=3-2\sqrt{2} x=2\sqrt{2}+3$$

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 2x,2.

$$1=2x\times 3-xx$$

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

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

Multiply 2 and 3 to get 6.

$$1=6x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$6x-x^{2}=1$$

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.

$$-x^{2}+6x=1$$

Divide both sides by -1.

$$\frac{-x^{2}+6x}{-1}=\frac{1}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{6}{-1}x=\frac{1}{-1}$$

Divide 6 by -1.

$$x^{2}-6x=\frac{1}{-1}$$

Divide 1 by -1.

$$x^{2}-6x=-1$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-1+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-1+9$$

Add -1 to 9.

$$x^{2}-6x+9=8$$

Factor x^{2}-6x+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}.

$$\left(x-3\right)^{2}=8$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{8}$$

Simplify.

$$x-3=2\sqrt{2} x-3=-2\sqrt{2}$$

Add 3 to both sides of the equation.

$$x=2\sqrt{2}+3 x=3-2\sqrt{2}$$