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

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

Solve for x

$$x = \frac{\sqrt{217} + 13}{8} \approx 3.466364983$$

$$x=\frac{13-\sqrt{217}}{8}\approx -0.216364983$$

Steps Using the Quadratic Formula

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

$$2x-\left(x-3\right)=4x\left(x-3\right)$$

To find the opposite of x-3, find the opposite of each term.

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

Combine 2x and -x to get x.

$$x+3=4x\left(x-3\right)$$

Use the distributive property to multiply 4x by x-3.

$$x+3=4x^{2}-12x$$

Subtract 4x^{2} from both sides.

$$x+3-4x^{2}=-12x$$

Add 12x to both sides.

$$x+3-4x^{2}+12x=0$$

Combine x and 12x to get 13x.

$$13x+3-4x^{2}=0$$

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.

$$-4x^{2}+13x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 13 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-4\right)\times 3}}{2\left(-4\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-4\right)\times 3}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-13±\sqrt{169+16\times 3}}{2\left(-4\right)}$$

Multiply 16 times 3.

$$x=\frac{-13±\sqrt{169+48}}{2\left(-4\right)}$$

Add 169 to 48.

$$x=\frac{-13±\sqrt{217}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-13±\sqrt{217}}{-8}$$

Now solve the equation x=\frac{-13±\sqrt{217}}{-8} when ± is plus. Add -13 to \sqrt{217}.

$$x=\frac{\sqrt{217}-13}{-8}$$

Divide -13+\sqrt{217} by -8.

$$x=\frac{13-\sqrt{217}}{8}$$

Now solve the equation x=\frac{-13±\sqrt{217}}{-8} when ± is minus. Subtract \sqrt{217} from -13.

$$x=\frac{-\sqrt{217}-13}{-8}$$

Divide -13-\sqrt{217} by -8.

$$x=\frac{\sqrt{217}+13}{8}$$

The equation is now solved.

$$x=\frac{13-\sqrt{217}}{8} x=\frac{\sqrt{217}+13}{8}$$

Steps for Completing the Square

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

$$2x-\left(x-3\right)=4x\left(x-3\right)$$

To find the opposite of x-3, find the opposite of each term.

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

Combine 2x and -x to get x.

$$x+3=4x\left(x-3\right)$$

Use the distributive property to multiply 4x by x-3.

$$x+3=4x^{2}-12x$$

Subtract 4x^{2} from both sides.

$$x+3-4x^{2}=-12x$$

Add 12x to both sides.

$$x+3-4x^{2}+12x=0$$

Combine x and 12x to get 13x.

$$13x+3-4x^{2}=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$13x-4x^{2}=-3$$

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.

$$-4x^{2}+13x=-3$$

Divide both sides by -4.

$$\frac{-4x^{2}+13x}{-4}=-\frac{3}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{13}{-4}x=-\frac{3}{-4}$$

Divide 13 by -4.

$$x^{2}-\frac{13}{4}x=-\frac{3}{-4}$$

Divide -3 by -4.

$$x^{2}-\frac{13}{4}x=\frac{3}{4}$$

Divide -\frac{13}{4}, the coefficient of the x term, by 2 to get -\frac{13}{8}. Then add the square of -\frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{4}x+\left(-\frac{13}{8}\right)^{2}=\frac{3}{4}+\left(-\frac{13}{8}\right)^{2}$$

Square -\frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=\frac{3}{4}+\frac{169}{64}$$

Add \frac{3}{4} to \frac{169}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=\frac{217}{64}$$

Factor x^{2}-\frac{13}{4}x+\frac{169}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{8}\right)^{2}=\frac{217}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{8}\right)^{2}}=\sqrt{\frac{217}{64}}$$

Simplify.

$$x-\frac{13}{8}=\frac{\sqrt{217}}{8} x-\frac{13}{8}=-\frac{\sqrt{217}}{8}$$

Add \frac{13}{8} to both sides of the equation.

$$x=\frac{\sqrt{217}+13}{8} x=\frac{13-\sqrt{217}}{8}$$