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

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

Solve for x

$$x = \frac{\sqrt{17} + 5}{4} \approx 2.280776406$$

$$x=\frac{5-\sqrt{17}}{4}\approx 0.219223594$$

Steps Using the Quadratic Formula

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

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

Combine 2x and x to get 3x.

$$3x-1=2x\left(x-1\right)$$

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

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

Subtract 2x^{2} from both sides.

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

Add 2x to both sides.

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

Combine 3x and 2x to get 5x.

$$5x-1-2x^{2}=0$$

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.

$$-2x^{2}+5x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 5 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-2\right)\left(-1\right)}}{2\left(-2\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-2\right)\left(-1\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-5±\sqrt{25+8\left(-1\right)}}{2\left(-2\right)}$$

Multiply 8 times -1.

$$x=\frac{-5±\sqrt{25-8}}{2\left(-2\right)}$$

Add 25 to -8.

$$x=\frac{-5±\sqrt{17}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-5±\sqrt{17}}{-4}$$

Now solve the equation x=\frac{-5±\sqrt{17}}{-4} when ± is plus. Add -5 to \sqrt{17}.

$$x=\frac{\sqrt{17}-5}{-4}$$

Divide -5+\sqrt{17} by -4.

$$x=\frac{5-\sqrt{17}}{4}$$

Now solve the equation x=\frac{-5±\sqrt{17}}{-4} when ± is minus. Subtract \sqrt{17} from -5.

$$x=\frac{-\sqrt{17}-5}{-4}$$

Divide -5-\sqrt{17} by -4.

$$x=\frac{\sqrt{17}+5}{4}$$

The equation is now solved.

$$x=\frac{5-\sqrt{17}}{4} x=\frac{\sqrt{17}+5}{4}$$

Steps for Completing the Square

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

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

Combine 2x and x to get 3x.

$$3x-1=2x\left(x-1\right)$$

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

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

Subtract 2x^{2} from both sides.

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

Add 2x to both sides.

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

Combine 3x and 2x to get 5x.

$$5x-1-2x^{2}=0$$

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

$$5x-2x^{2}=1$$

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.

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

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

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

Divide 5 by -2.

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

Divide 1 by -2.

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

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=-\frac{1}{2}+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=-\frac{1}{2}+\frac{25}{16}$$

Add -\frac{1}{2} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{17}{16}$$

Factor x^{2}-\frac{5}{2}x+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{4}\right)^{2}=\frac{17}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{\frac{17}{16}}$$

Simplify.

$$x-\frac{5}{4}=\frac{\sqrt{17}}{4} x-\frac{5}{4}=-\frac{\sqrt{17}}{4}$$

Add \frac{5}{4} to both sides of the equation.

$$x=\frac{\sqrt{17}+5}{4} x=\frac{5-\sqrt{17}}{4}$$