Solve 1/x+1/x-14=1/50

$$\frac{ 1 }{ x } + \frac{ 1 }{ x-14 } = \frac{ 1 }{ 50 }$$

Solve for x

$$x=\sqrt{2549}+57\approx 107.487622245$$

$$x=57-\sqrt{2549}\approx 6.512377755$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,14 since division by zero is not defined. Multiply both sides of the equation by 50x\left(x-14\right), the least common multiple of x,x-14,50.

$$50x-700+50x=x\left(x-14\right)$$

Combine 50x and 50x to get 100x.

$$100x-700=x\left(x-14\right)$$

Use the distributive property to multiply x by x-14.

$$100x-700=x^{2}-14x$$

Subtract x^{2} from both sides.

$$100x-700-x^{2}=-14x$$

Add 14x to both sides.

$$100x-700-x^{2}+14x=0$$

Combine 100x and 14x to get 114x.

$$114x-700-x^{2}=0$$

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.

$$-x^{2}+114x-700=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 114 for b, and -700 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-114±\sqrt{114^{2}-4\left(-1\right)\left(-700\right)}}{2\left(-1\right)}$$

Square 114.

$$x=\frac{-114±\sqrt{12996-4\left(-1\right)\left(-700\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-114±\sqrt{12996+4\left(-700\right)}}{2\left(-1\right)}$$

Multiply 4 times -700.

$$x=\frac{-114±\sqrt{12996-2800}}{2\left(-1\right)}$$

Add 12996 to -2800.

$$x=\frac{-114±\sqrt{10196}}{2\left(-1\right)}$$

Take the square root of 10196.

$$x=\frac{-114±2\sqrt{2549}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-114±2\sqrt{2549}}{-2}$$

Now solve the equation x=\frac{-114±2\sqrt{2549}}{-2} when ± is plus. Add -114 to 2\sqrt{2549}.

$$x=\frac{2\sqrt{2549}-114}{-2}$$

Divide -114+2\sqrt{2549} by -2.

$$x=57-\sqrt{2549}$$

Now solve the equation x=\frac{-114±2\sqrt{2549}}{-2} when ± is minus. Subtract 2\sqrt{2549} from -114.

$$x=\frac{-2\sqrt{2549}-114}{-2}$$

Divide -114-2\sqrt{2549} by -2.

$$x=\sqrt{2549}+57$$

The equation is now solved.

$$x=57-\sqrt{2549} x=\sqrt{2549}+57$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,14 since division by zero is not defined. Multiply both sides of the equation by 50x\left(x-14\right), the least common multiple of x,x-14,50.

$$50x-700+50x=x\left(x-14\right)$$

Combine 50x and 50x to get 100x.

$$100x-700=x\left(x-14\right)$$

Use the distributive property to multiply x by x-14.

$$100x-700=x^{2}-14x$$

Subtract x^{2} from both sides.

$$100x-700-x^{2}=-14x$$

Add 14x to both sides.

$$100x-700-x^{2}+14x=0$$

Combine 100x and 14x to get 114x.

$$114x-700-x^{2}=0$$

Add 700 to both sides. Anything plus zero gives itself.

$$114x-x^{2}=700$$

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.

$$-x^{2}+114x=700$$

Divide both sides by -1.

$$\frac{-x^{2}+114x}{-1}=\frac{700}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{114}{-1}x=\frac{700}{-1}$$

Divide 114 by -1.

$$x^{2}-114x=\frac{700}{-1}$$

Divide 700 by -1.

$$x^{2}-114x=-700$$

Divide -114, the coefficient of the x term, by 2 to get -57. Then add the square of -57 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-114x+\left(-57\right)^{2}=-700+\left(-57\right)^{2}$$

Square -57.

$$x^{2}-114x+3249=-700+3249$$

Add -700 to 3249.

$$x^{2}-114x+3249=2549$$

Factor x^{2}-114x+3249. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-57\right)^{2}=2549$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-57\right)^{2}}=\sqrt{2549}$$

Simplify.

$$x-57=\sqrt{2549} x-57=-\sqrt{2549}$$

Add 57 to both sides of the equation.

$$x=\sqrt{2549}+57 x=57-\sqrt{2549}$$