Solve 1/x+1/x+10=1/24

$$\frac { 1 } { x } + \frac { 1 } { x + 10 } = \frac { 1 } { 24 }$$

Solve for x

$$x=\sqrt{601}+19\approx 43.515301344$$

$$x=19-\sqrt{601}\approx -5.515301344$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 24x\left(x+10\right), the least common multiple of x,x+10,24.

$$24x+240+24x=x\left(x+10\right)$$

Combine 24x and 24x to get 48x.

$$48x+240=x\left(x+10\right)$$

Use the distributive property to multiply x by x+10.

$$48x+240=x^{2}+10x$$

Subtract x^{2} from both sides.

$$48x+240-x^{2}=10x$$

Subtract 10x from both sides.

$$48x+240-x^{2}-10x=0$$

Combine 48x and -10x to get 38x.

$$38x+240-x^{2}=0$$

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.

$$-x^{2}+38x+240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 38 for b, and 240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-38±\sqrt{38^{2}-4\left(-1\right)\times 240}}{2\left(-1\right)}$$

Square 38.

$$x=\frac{-38±\sqrt{1444-4\left(-1\right)\times 240}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-38±\sqrt{1444+4\times 240}}{2\left(-1\right)}$$

Multiply 4 times 240.

$$x=\frac{-38±\sqrt{1444+960}}{2\left(-1\right)}$$

Add 1444 to 960.

$$x=\frac{-38±\sqrt{2404}}{2\left(-1\right)}$$

Take the square root of 2404.

$$x=\frac{-38±2\sqrt{601}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-38±2\sqrt{601}}{-2}$$

Now solve the equation x=\frac{-38±2\sqrt{601}}{-2} when ± is plus. Add -38 to 2\sqrt{601}.

$$x=\frac{2\sqrt{601}-38}{-2}$$

Divide -38+2\sqrt{601} by -2.

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

Now solve the equation x=\frac{-38±2\sqrt{601}}{-2} when ± is minus. Subtract 2\sqrt{601} from -38.

$$x=\frac{-2\sqrt{601}-38}{-2}$$

Divide -38-2\sqrt{601} by -2.

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

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 24x\left(x+10\right), the least common multiple of x,x+10,24.

$$24x+240+24x=x\left(x+10\right)$$

Combine 24x and 24x to get 48x.

$$48x+240=x\left(x+10\right)$$

Use the distributive property to multiply x by x+10.

$$48x+240=x^{2}+10x$$

Subtract x^{2} from both sides.

$$48x+240-x^{2}=10x$$

Subtract 10x from both sides.

$$48x+240-x^{2}-10x=0$$

Combine 48x and -10x to get 38x.

$$38x+240-x^{2}=0$$

Subtract 240 from both sides. Anything subtracted from zero gives its negation.

$$38x-x^{2}=-240$$

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.

$$-x^{2}+38x=-240$$

Divide both sides by -1.

$$\frac{-x^{2}+38x}{-1}=-\frac{240}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{38}{-1}x=-\frac{240}{-1}$$

Divide 38 by -1.

$$x^{2}-38x=-\frac{240}{-1}$$

Divide -240 by -1.

$$x^{2}-38x=240$$

Divide -38, the coefficient of the x term, by 2 to get -19. Then add the square of -19 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-38x+\left(-19\right)^{2}=240+\left(-19\right)^{2}$$

Square -19.

$$x^{2}-38x+361=240+361$$

Add 240 to 361.

$$x^{2}-38x+361=601$$

Factor x^{2}-38x+361. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 19 to both sides of the equation.

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