Solve 1/x-4+1/x=2

$$\frac { 1 } { x - 4 } + \frac { 1 } { x } = 2$$

Solve for x

$$x = \frac{\sqrt{17} + 5}{2} \approx 4.561552813$$

$$x=\frac{5-\sqrt{17}}{2}\approx 0.438447187$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,4 since division by zero is not defined. Multiply both sides of the equation by x\left(x-4\right), the least common multiple of x-4,x.

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

Combine x and x to get 2x.

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

Use the distributive property to multiply 2x by x-4.

$$2x-4=2x^{2}-8x$$

Subtract 2x^{2} from both sides.

$$2x-4-2x^{2}=-8x$$

Add 8x to both sides.

$$2x-4-2x^{2}+8x=0$$

Combine 2x and 8x to get 10x.

$$10x-4-2x^{2}=0$$

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.

$$-2x^{2}+10x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 10 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-2\right)\left(-4\right)}}{2\left(-2\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-2\right)\left(-4\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-10±\sqrt{100+8\left(-4\right)}}{2\left(-2\right)}$$

Multiply 8 times -4.

$$x=\frac{-10±\sqrt{100-32}}{2\left(-2\right)}$$

Add 100 to -32.

$$x=\frac{-10±\sqrt{68}}{2\left(-2\right)}$$

Take the square root of 68.

$$x=\frac{-10±2\sqrt{17}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-10±2\sqrt{17}}{-4}$$

Now solve the equation x=\frac{-10±2\sqrt{17}}{-4} when ± is plus. Add -10 to 2\sqrt{17}.

$$x=\frac{2\sqrt{17}-10}{-4}$$

Divide -10+2\sqrt{17} by -4.

$$x=\frac{5-\sqrt{17}}{2}$$

Now solve the equation x=\frac{-10±2\sqrt{17}}{-4} when ± is minus. Subtract 2\sqrt{17} from -10.

$$x=\frac{-2\sqrt{17}-10}{-4}$$

Divide -10-2\sqrt{17} by -4.

$$x=\frac{\sqrt{17}+5}{2}$$

The equation is now solved.

$$x=\frac{5-\sqrt{17}}{2} x=\frac{\sqrt{17}+5}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,4 since division by zero is not defined. Multiply both sides of the equation by x\left(x-4\right), the least common multiple of x-4,x.

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

Combine x and x to get 2x.

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

Use the distributive property to multiply 2x by x-4.

$$2x-4=2x^{2}-8x$$

Subtract 2x^{2} from both sides.

$$2x-4-2x^{2}=-8x$$

Add 8x to both sides.

$$2x-4-2x^{2}+8x=0$$

Combine 2x and 8x to get 10x.

$$10x-4-2x^{2}=0$$

Add 4 to both sides. Anything plus zero gives itself.

$$10x-2x^{2}=4$$

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.

$$-2x^{2}+10x=4$$

Divide both sides by -2.

$$\frac{-2x^{2}+10x}{-2}=\frac{4}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{10}{-2}x=\frac{4}{-2}$$

Divide 10 by -2.

$$x^{2}-5x=\frac{4}{-2}$$

Divide 4 by -2.

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

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

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

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-2+\frac{25}{4}$$

Add -2 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{17}{4}$$

Factor x^{2}-5x+\frac{25}{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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{17}}{2} x-\frac{5}{2}=-\frac{\sqrt{17}}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{\sqrt{17}+5}{2} x=\frac{5-\sqrt{17}}{2}$$