Solve 1/x+1/x+2=17/72

$$\frac{ 1 }{ x } + \frac{ 1 }{ x+2 } = \frac{ 17 }{ 72 }$$

Solve for x

$$x = \frac{\sqrt{5473} + 55}{17} \approx 7.587042762$$

$$x=\frac{55-\sqrt{5473}}{17}\approx -1.116454527$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 72x\left(x+2\right), the least common multiple of x,x+2,72.

$$72x+144+72x=17x\left(x+2\right)$$

Combine 72x and 72x to get 144x.

$$144x+144=17x\left(x+2\right)$$

Use the distributive property to multiply 17x by x+2.

$$144x+144=17x^{2}+34x$$

Subtract 17x^{2} from both sides.

$$144x+144-17x^{2}=34x$$

Subtract 34x from both sides.

$$144x+144-17x^{2}-34x=0$$

Combine 144x and -34x to get 110x.

$$110x+144-17x^{2}=0$$

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.

$$-17x^{2}+110x+144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -17 for a, 110 for b, and 144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-110±\sqrt{110^{2}-4\left(-17\right)\times 144}}{2\left(-17\right)}$$

Square 110.

$$x=\frac{-110±\sqrt{12100-4\left(-17\right)\times 144}}{2\left(-17\right)}$$

Multiply -4 times -17.

$$x=\frac{-110±\sqrt{12100+68\times 144}}{2\left(-17\right)}$$

Multiply 68 times 144.

$$x=\frac{-110±\sqrt{12100+9792}}{2\left(-17\right)}$$

Add 12100 to 9792.

$$x=\frac{-110±\sqrt{21892}}{2\left(-17\right)}$$

Take the square root of 21892.

$$x=\frac{-110±2\sqrt{5473}}{2\left(-17\right)}$$

Multiply 2 times -17.

$$x=\frac{-110±2\sqrt{5473}}{-34}$$

Now solve the equation x=\frac{-110±2\sqrt{5473}}{-34} when ± is plus. Add -110 to 2\sqrt{5473}.

$$x=\frac{2\sqrt{5473}-110}{-34}$$

Divide -110+2\sqrt{5473} by -34.

$$x=\frac{55-\sqrt{5473}}{17}$$

Now solve the equation x=\frac{-110±2\sqrt{5473}}{-34} when ± is minus. Subtract 2\sqrt{5473} from -110.

$$x=\frac{-2\sqrt{5473}-110}{-34}$$

Divide -110-2\sqrt{5473} by -34.

$$x=\frac{\sqrt{5473}+55}{17}$$

The equation is now solved.

$$x=\frac{55-\sqrt{5473}}{17} x=\frac{\sqrt{5473}+55}{17}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 72x\left(x+2\right), the least common multiple of x,x+2,72.

$$72x+144+72x=17x\left(x+2\right)$$

Combine 72x and 72x to get 144x.

$$144x+144=17x\left(x+2\right)$$

Use the distributive property to multiply 17x by x+2.

$$144x+144=17x^{2}+34x$$

Subtract 17x^{2} from both sides.

$$144x+144-17x^{2}=34x$$

Subtract 34x from both sides.

$$144x+144-17x^{2}-34x=0$$

Combine 144x and -34x to get 110x.

$$110x+144-17x^{2}=0$$

Subtract 144 from both sides. Anything subtracted from zero gives its negation.

$$110x-17x^{2}=-144$$

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.

$$-17x^{2}+110x=-144$$

Divide both sides by -17.

$$\frac{-17x^{2}+110x}{-17}=-\frac{144}{-17}$$

Dividing by -17 undoes the multiplication by -17.

$$x^{2}+\frac{110}{-17}x=-\frac{144}{-17}$$

Divide 110 by -17.

$$x^{2}-\frac{110}{17}x=-\frac{144}{-17}$$

Divide -144 by -17.

$$x^{2}-\frac{110}{17}x=\frac{144}{17}$$

Divide -\frac{110}{17}, the coefficient of the x term, by 2 to get -\frac{55}{17}. Then add the square of -\frac{55}{17} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{110}{17}x+\left(-\frac{55}{17}\right)^{2}=\frac{144}{17}+\left(-\frac{55}{17}\right)^{2}$$

Square -\frac{55}{17} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{110}{17}x+\frac{3025}{289}=\frac{144}{17}+\frac{3025}{289}$$

Add \frac{144}{17} to \frac{3025}{289} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{110}{17}x+\frac{3025}{289}=\frac{5473}{289}$$

Factor x^{2}-\frac{110}{17}x+\frac{3025}{289}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{55}{17}\right)^{2}=\frac{5473}{289}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{55}{17}\right)^{2}}=\sqrt{\frac{5473}{289}}$$

Simplify.

$$x-\frac{55}{17}=\frac{\sqrt{5473}}{17} x-\frac{55}{17}=-\frac{\sqrt{5473}}{17}$$

Add \frac{55}{17} to both sides of the equation.

$$x=\frac{\sqrt{5473}+55}{17} x=\frac{55-\sqrt{5473}}{17}$$