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

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

Solve for x

$$x=\frac{\sqrt{2}}{2}\approx 0.707106781$$

$$x=-\frac{\sqrt{2}}{2}\approx -0.707106781$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values 0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by x\left(2x-1\right), the least common multiple of 2x-1,x.

$$x-\left(2x-1\right)=x\left(2x-1\right)$$

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

$$x-2x+1=x\left(2x-1\right)$$

Combine x and -2x to get -x.

$$-x+1=x\left(2x-1\right)$$

Use the distributive property to multiply x by 2x-1.

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

Subtract 2x^{2} from both sides.

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

Add x to both sides.

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

Combine -x and x to get 0.

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

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by -2.

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

Fraction \frac{-1}{-2} can be simplified to \frac{1}{2} by removing the negative sign from both the numerator and the denominator.

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

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by x\left(2x-1\right), the least common multiple of 2x-1,x.

$$x-\left(2x-1\right)=x\left(2x-1\right)$$

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

$$x-2x+1=x\left(2x-1\right)$$

Combine x and -2x to get -x.

$$-x+1=x\left(2x-1\right)$$

Use the distributive property to multiply x by 2x-1.

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

Subtract 2x^{2} from both sides.

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

Add x to both sides.

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

Combine -x and x to get 0.

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

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

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

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

$$x=\frac{0±\sqrt{0^{2}-4\left(-2\right)}}{2\left(-2\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-2\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{0±\sqrt{8}}{2\left(-2\right)}$$

Take the square root of 8.

$$x=\frac{0±2\sqrt{2}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{0±2\sqrt{2}}{-4}$$

Now solve the equation x=\frac{0±2\sqrt{2}}{-4} when ± is plus.

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

Now solve the equation x=\frac{0±2\sqrt{2}}{-4} when ± is minus.

$$x=\frac{\sqrt{2}}{2}$$

The equation is now solved.

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