Solve 1/x+1/x+5=1/12

$$\frac { 1 } { x } + \frac { 1 } { x + 5 } = \frac { 1 } { 12 }$$

Solve for x

$$x = \frac{\sqrt{601} + 19}{2} \approx 21.757650672$$

$$x=\frac{19-\sqrt{601}}{2}\approx -2.757650672$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,0 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x+5\right), the least common multiple of x,x+5,12.

$$12x+60+12x=x\left(x+5\right)$$

Combine 12x and 12x to get 24x.

$$24x+60=x\left(x+5\right)$$

Use the distributive property to multiply x by x+5.

$$24x+60=x^{2}+5x$$

Subtract x^{2} from both sides.

$$24x+60-x^{2}=5x$$

Subtract 5x from both sides.

$$24x+60-x^{2}-5x=0$$

Combine 24x and -5x to get 19x.

$$19x+60-x^{2}=0$$

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.

$$-x^{2}+19x+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 19 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-1\right)\times 60}}{2\left(-1\right)}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\left(-1\right)\times 60}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-19±\sqrt{361+4\times 60}}{2\left(-1\right)}$$

Multiply 4 times 60.

$$x=\frac{-19±\sqrt{361+240}}{2\left(-1\right)}$$

Add 361 to 240.

$$x=\frac{-19±\sqrt{601}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-19±\sqrt{601}}{-2}$$

Now solve the equation x=\frac{-19±\sqrt{601}}{-2} when ± is plus. Add -19 to \sqrt{601}.

$$x=\frac{\sqrt{601}-19}{-2}$$

Divide -19+\sqrt{601} by -2.

$$x=\frac{19-\sqrt{601}}{2}$$

Now solve the equation x=\frac{-19±\sqrt{601}}{-2} when ± is minus. Subtract \sqrt{601} from -19.

$$x=\frac{-\sqrt{601}-19}{-2}$$

Divide -19-\sqrt{601} by -2.

$$x=\frac{\sqrt{601}+19}{2}$$

The equation is now solved.

$$x=\frac{19-\sqrt{601}}{2} x=\frac{\sqrt{601}+19}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,0 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x+5\right), the least common multiple of x,x+5,12.

$$12x+60+12x=x\left(x+5\right)$$

Combine 12x and 12x to get 24x.

$$24x+60=x\left(x+5\right)$$

Use the distributive property to multiply x by x+5.

$$24x+60=x^{2}+5x$$

Subtract x^{2} from both sides.

$$24x+60-x^{2}=5x$$

Subtract 5x from both sides.

$$24x+60-x^{2}-5x=0$$

Combine 24x and -5x to get 19x.

$$19x+60-x^{2}=0$$

Subtract 60 from both sides. Anything subtracted from zero gives its negation.

$$19x-x^{2}=-60$$

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.

$$-x^{2}+19x=-60$$

Divide both sides by -1.

$$\frac{-x^{2}+19x}{-1}=-\frac{60}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{19}{-1}x=-\frac{60}{-1}$$

Divide 19 by -1.

$$x^{2}-19x=-\frac{60}{-1}$$

Divide -60 by -1.

$$x^{2}-19x=60$$

Divide -19, the coefficient of the x term, by 2 to get -\frac{19}{2}. Then add the square of -\frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-19x+\left(-\frac{19}{2}\right)^{2}=60+\left(-\frac{19}{2}\right)^{2}$$

Square -\frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-19x+\frac{361}{4}=60+\frac{361}{4}$$

Add 60 to \frac{361}{4}.

$$x^{2}-19x+\frac{361}{4}=\frac{601}{4}$$

Factor x^{2}-19x+\frac{361}{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}.

$$\left(x-\frac{19}{2}\right)^{2}=\frac{601}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{2}\right)^{2}}=\sqrt{\frac{601}{4}}$$

Simplify.

$$x-\frac{19}{2}=\frac{\sqrt{601}}{2} x-\frac{19}{2}=-\frac{\sqrt{601}}{2}$$

Add \frac{19}{2} to both sides of the equation.

$$x=\frac{\sqrt{601}+19}{2} x=\frac{19-\sqrt{601}}{2}$$