Solve 1/x+1/x-10=2/25

$$\frac { 1 } { x } + \frac { 1 } { x - 10 } = \frac { 2 } { 25 }$$

Solve for x

$$x = \frac{5 \sqrt{29} + 35}{2} \approx 30.962912018$$

$$x = \frac{35 - 5 \sqrt{29}}{2} \approx 4.037087982$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,10 since division by zero is not defined. Multiply both sides of the equation by 25x\left(x-10\right), the least common multiple of x,x-10,25.

$$25x-250+25x=2x\left(x-10\right)$$

Combine 25x and 25x to get 50x.

$$50x-250=2x\left(x-10\right)$$

Use the distributive property to multiply 2x by x-10.

$$50x-250=2x^{2}-20x$$

Subtract 2x^{2} from both sides.

$$50x-250-2x^{2}=-20x$$

Add 20x to both sides.

$$50x-250-2x^{2}+20x=0$$

Combine 50x and 20x to get 70x.

$$70x-250-2x^{2}=0$$

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.

$$-2x^{2}+70x-250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 70 for b, and -250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-70±\sqrt{70^{2}-4\left(-2\right)\left(-250\right)}}{2\left(-2\right)}$$

Square 70.

$$x=\frac{-70±\sqrt{4900-4\left(-2\right)\left(-250\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-70±\sqrt{4900+8\left(-250\right)}}{2\left(-2\right)}$$

Multiply 8 times -250.

$$x=\frac{-70±\sqrt{4900-2000}}{2\left(-2\right)}$$

Add 4900 to -2000.

$$x=\frac{-70±\sqrt{2900}}{2\left(-2\right)}$$

Take the square root of 2900.

$$x=\frac{-70±10\sqrt{29}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-70±10\sqrt{29}}{-4}$$

Now solve the equation x=\frac{-70±10\sqrt{29}}{-4} when ± is plus. Add -70 to 10\sqrt{29}.

$$x=\frac{10\sqrt{29}-70}{-4}$$

Divide -70+10\sqrt{29} by -4.

$$x=\frac{35-5\sqrt{29}}{2}$$

Now solve the equation x=\frac{-70±10\sqrt{29}}{-4} when ± is minus. Subtract 10\sqrt{29} from -70.

$$x=\frac{-10\sqrt{29}-70}{-4}$$

Divide -70-10\sqrt{29} by -4.

$$x=\frac{5\sqrt{29}+35}{2}$$

The equation is now solved.

$$x=\frac{35-5\sqrt{29}}{2} x=\frac{5\sqrt{29}+35}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,10 since division by zero is not defined. Multiply both sides of the equation by 25x\left(x-10\right), the least common multiple of x,x-10,25.

$$25x-250+25x=2x\left(x-10\right)$$

Combine 25x and 25x to get 50x.

$$50x-250=2x\left(x-10\right)$$

Use the distributive property to multiply 2x by x-10.

$$50x-250=2x^{2}-20x$$

Subtract 2x^{2} from both sides.

$$50x-250-2x^{2}=-20x$$

Add 20x to both sides.

$$50x-250-2x^{2}+20x=0$$

Combine 50x and 20x to get 70x.

$$70x-250-2x^{2}=0$$

Add 250 to both sides. Anything plus zero gives itself.

$$70x-2x^{2}=250$$

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.

$$-2x^{2}+70x=250$$

Divide both sides by -2.

$$\frac{-2x^{2}+70x}{-2}=\frac{250}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{70}{-2}x=\frac{250}{-2}$$

Divide 70 by -2.

$$x^{2}-35x=\frac{250}{-2}$$

Divide 250 by -2.

$$x^{2}-35x=-125$$

Divide -35, the coefficient of the x term, by 2 to get -\frac{35}{2}. Then add the square of -\frac{35}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-35x+\left(-\frac{35}{2}\right)^{2}=-125+\left(-\frac{35}{2}\right)^{2}$$

Square -\frac{35}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-35x+\frac{1225}{4}=-125+\frac{1225}{4}$$

Add -125 to \frac{1225}{4}.

$$x^{2}-35x+\frac{1225}{4}=\frac{725}{4}$$

Factor x^{2}-35x+\frac{1225}{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}.

$$\left(x-\frac{35}{2}\right)^{2}=\frac{725}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{2}\right)^{2}}=\sqrt{\frac{725}{4}}$$

Simplify.

$$x-\frac{35}{2}=\frac{5\sqrt{29}}{2} x-\frac{35}{2}=-\frac{5\sqrt{29}}{2}$$

Add \frac{35}{2} to both sides of the equation.

$$x=\frac{5\sqrt{29}+35}{2} x=\frac{35-5\sqrt{29}}{2}$$