Solve (1/xdiv2)+1/x=1/16

$$( \frac{ 1 }{ x } \div 2)+ \frac{ 1 }{ x } = \frac{ 1 }{ 16 }$$

Solve for x

$$x=24$$

Steps Using Factoring

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 16x, the least common multiple of 2,x,16.

$$8x\times \frac{1}{x}+16=x$$

Express 8\times \frac{1}{x} as a single fraction.

$$\frac{8}{x}x+16=x$$

Express \frac{8}{x}x as a single fraction.

$$\frac{8x}{x}+16=x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 16 times \frac{x}{x}.

$$\frac{8x}{x}+\frac{16x}{x}=x$$

Since \frac{8x}{x} and \frac{16x}{x} have the same denominator, add them by adding their numerators.

$$\frac{8x+16x}{x}=x$$

Combine like terms in 8x+16x.

$$\frac{24x}{x}=x$$

Subtract x from both sides.

$$\frac{24x}{x}-x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.

$$\frac{24x}{x}-\frac{xx}{x}=0$$

Since \frac{24x}{x} and \frac{xx}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{24x-xx}{x}=0$$

Do the multiplications in 24x-xx.

$$\frac{24x-x^{2}}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$24x-x^{2}=0$$

Factor out x.

$$x\left(24-x\right)=0$$

To find equation solutions, solve x=0 and 24-x=0.

$$x=0 x=24$$

Variable x cannot be equal to 0.

$$x=24$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 16x, the least common multiple of 2,x,16.

$$8x\times \frac{1}{x}+16=x$$

Express 8\times \frac{1}{x} as a single fraction.

$$\frac{8}{x}x+16=x$$

Express \frac{8}{x}x as a single fraction.

$$\frac{8x}{x}+16=x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 16 times \frac{x}{x}.

$$\frac{8x}{x}+\frac{16x}{x}=x$$

Since \frac{8x}{x} and \frac{16x}{x} have the same denominator, add them by adding their numerators.

$$\frac{8x+16x}{x}=x$$

Combine like terms in 8x+16x.

$$\frac{24x}{x}=x$$

Subtract x from both sides.

$$\frac{24x}{x}-x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.

$$\frac{24x}{x}-\frac{xx}{x}=0$$

Since \frac{24x}{x} and \frac{xx}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{24x-xx}{x}=0$$

Do the multiplications in 24x-xx.

$$\frac{24x-x^{2}}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$24x-x^{2}=0$$

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 24 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}}}{2\left(-1\right)}$$

Take the square root of 24^{2}.

$$x=\frac{-24±24}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-24±24}{-2}$$

Now solve the equation x=\frac{-24±24}{-2} when ± is plus. Add -24 to 24.

$$x=\frac{0}{-2}$$

Divide 0 by -2.

$$x=0$$

Now solve the equation x=\frac{-24±24}{-2} when ± is minus. Subtract 24 from -24.

$$x=-\frac{48}{-2}$$

Divide -48 by -2.

$$x=24$$

The equation is now solved.

$$x=0 x=24$$

Variable x cannot be equal to 0.

$$x=24$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 16x, the least common multiple of 2,x,16.

$$8x\times \frac{1}{x}+16=x$$

Express 8\times \frac{1}{x} as a single fraction.

$$\frac{8}{x}x+16=x$$

Express \frac{8}{x}x as a single fraction.

$$\frac{8x}{x}+16=x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 16 times \frac{x}{x}.

$$\frac{8x}{x}+\frac{16x}{x}=x$$

Since \frac{8x}{x} and \frac{16x}{x} have the same denominator, add them by adding their numerators.

$$\frac{8x+16x}{x}=x$$

Combine like terms in 8x+16x.

$$\frac{24x}{x}=x$$

Subtract x from both sides.

$$\frac{24x}{x}-x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.

$$\frac{24x}{x}-\frac{xx}{x}=0$$

Since \frac{24x}{x} and \frac{xx}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{24x-xx}{x}=0$$

Do the multiplications in 24x-xx.

$$\frac{24x-x^{2}}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$24x-x^{2}=0$$

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.

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

Divide both sides by -1.

$$\frac{-x^{2}+24x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{24}{-1}x=\frac{0}{-1}$$

Divide 24 by -1.

$$x^{2}-24x=\frac{0}{-1}$$

Divide 0 by -1.

$$x^{2}-24x=0$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24x+\left(-12\right)^{2}=\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=144$$

Factor x^{2}-24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12\right)^{2}=144$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{144}$$

Simplify.

$$x-12=12 x-12=-12$$

Add 12 to both sides of the equation.

$$x=24 x=0$$

Variable x cannot be equal to 0.

$$x=24$$