Solve x/2-1-2x/2x=5x

$$\frac { x } { 2 } - \frac { 1 - 2 x } { 2 x } = 5 x$$

Solve for x (complex solution)

$$x=\frac{-2\sqrt{2}i+1}{9}\approx 0.111111111-0.314269681i$$

$$x=\frac{1+2\sqrt{2}i}{9}\approx 0.111111111+0.314269681i$$

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.

$$xx-\left(1-2x\right)=5x\times 2x$$

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

$$x^{2}-\left(1-2x\right)=5x\times 2x$$

To find the opposite of 1-2x, find the opposite of each term.

$$x^{2}-1+2x=5x\times 2x$$

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

$$x^{2}-1+2x=5x^{2}\times 2$$

Multiply 5 and 2 to get 10.

$$x^{2}-1+2x=10x^{2}$$

Subtract 10x^{2} from both sides.

$$x^{2}-1+2x-10x^{2}=0$$

Combine x^{2} and -10x^{2} to get -9x^{2}.

$$-9x^{2}-1+2x=0$$

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.

$$-9x^{2}+2x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 2 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-9\right)\left(-1\right)}}{2\left(-9\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-9\right)\left(-1\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-2±\sqrt{4+36\left(-1\right)}}{2\left(-9\right)}$$

Multiply 36 times -1.

$$x=\frac{-2±\sqrt{4-36}}{2\left(-9\right)}$$

Add 4 to -36.

$$x=\frac{-2±\sqrt{-32}}{2\left(-9\right)}$$

Take the square root of -32.

$$x=\frac{-2±4\sqrt{2}i}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-2±4\sqrt{2}i}{-18}$$

Now solve the equation x=\frac{-2±4\sqrt{2}i}{-18} when ± is plus. Add -2 to 4i\sqrt{2}.

$$x=\frac{-2+4\sqrt{2}i}{-18}$$

Divide -2+4i\sqrt{2} by -18.

$$x=\frac{-2\sqrt{2}i+1}{9}$$

Now solve the equation x=\frac{-2±4\sqrt{2}i}{-18} when ± is minus. Subtract 4i\sqrt{2} from -2.

$$x=\frac{-4\sqrt{2}i-2}{-18}$$

Divide -2-4i\sqrt{2} by -18.

$$x=\frac{1+2\sqrt{2}i}{9}$$

The equation is now solved.

$$x=\frac{-2\sqrt{2}i+1}{9} x=\frac{1+2\sqrt{2}i}{9}$$

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.

$$xx-\left(1-2x\right)=5x\times 2x$$

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

$$x^{2}-\left(1-2x\right)=5x\times 2x$$

To find the opposite of 1-2x, find the opposite of each term.

$$x^{2}-1+2x=5x\times 2x$$

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

$$x^{2}-1+2x=5x^{2}\times 2$$

Multiply 5 and 2 to get 10.

$$x^{2}-1+2x=10x^{2}$$

Subtract 10x^{2} from both sides.

$$x^{2}-1+2x-10x^{2}=0$$

Combine x^{2} and -10x^{2} to get -9x^{2}.

$$-9x^{2}-1+2x=0$$

Add 1 to both sides. Anything plus zero gives itself.

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

Divide both sides by -9.

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

Dividing by -9 undoes the multiplication by -9.

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

Divide 2 by -9.

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

Divide 1 by -9.

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

Divide -\frac{2}{9}, the coefficient of the x term, by 2 to get -\frac{1}{9}. Then add the square of -\frac{1}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{9}x+\left(-\frac{1}{9}\right)^{2}=-\frac{1}{9}+\left(-\frac{1}{9}\right)^{2}$$

Square -\frac{1}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{9}x+\frac{1}{81}=-\frac{1}{9}+\frac{1}{81}$$

Add -\frac{1}{9} to \frac{1}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2}{9}x+\frac{1}{81}=-\frac{8}{81}$$

Factor x^{2}-\frac{2}{9}x+\frac{1}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{9}\right)^{2}=-\frac{8}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{9}\right)^{2}}=\sqrt{-\frac{8}{81}}$$

Simplify.

$$x-\frac{1}{9}=\frac{2\sqrt{2}i}{9} x-\frac{1}{9}=-\frac{2\sqrt{2}i}{9}$$

Add \frac{1}{9} to both sides of the equation.

$$x=\frac{1+2\sqrt{2}i}{9} x=\frac{-2\sqrt{2}i+1}{9}$$