Solve x-2/2x-3=1/x

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

Solve for x

$$x=1$$

$$x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by x\left(2x-3\right), the least common multiple of 2x-3,x.

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

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

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

Subtract 2x from both sides.

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

Combine -2x and -2x to get -4x.

$$x^{2}-4x=-3$$

Add 3 to both sides.

$$x^{2}-4x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 3}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 3}}{2}$$

Multiply -4 times 3.

$$x=\frac{-\left(-4\right)±\sqrt{16-12}}{2}$$

Add 16 to -12.

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

Take the square root of 4.

$$x=\frac{-\left(-4\right)±2}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±2}{2}$$

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

$$x=\frac{6}{2}$$

Divide 6 by 2.

$$x=3$$

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

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

Divide 2 by 2.

$$x=1$$

The equation is now solved.

$$x=3 x=1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by x\left(2x-3\right), the least common multiple of 2x-3,x.

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

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

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

Subtract 2x from both sides.

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

Combine -2x and -2x to get -4x.

$$x^{2}-4x=-3$$

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

$$x^{2}-4x+\left(-2\right)^{2}=-3+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-3+4$$

Add -3 to 4.

$$x^{2}-4x+4=1$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-2=1 x-2=-1$$

Add 2 to both sides of the equation.

$$x=3 x=1$$