Solve 1/x+1/100-x=1/23

$$\frac{ 1 }{ x } + \frac{ 1 }{ 100-x } = \frac{ 1 }{ 23 }$$

Solve for x

$$x=10\sqrt{2}+50\approx 64.142135624$$

$$x=50-10\sqrt{2}\approx 35.857864376$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,100 since division by zero is not defined. Multiply both sides of the equation by 23x\left(x-100\right), the least common multiple of x,100-x,23.

$$23x-2300-23x=x\left(x-100\right)$$

Combine 23x and -23x to get 0.

$$-2300=x\left(x-100\right)$$

Use the distributive property to multiply x by x-100.

$$-2300=x^{2}-100x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-100x=-2300$$

Add 2300 to both sides.

$$x^{2}-100x+2300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -100 for b, and 2300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-100\right)±\sqrt{\left(-100\right)^{2}-4\times 2300}}{2}$$

Square -100.

$$x=\frac{-\left(-100\right)±\sqrt{10000-4\times 2300}}{2}$$

Multiply -4 times 2300.

$$x=\frac{-\left(-100\right)±\sqrt{10000-9200}}{2}$$

Add 10000 to -9200.

$$x=\frac{-\left(-100\right)±\sqrt{800}}{2}$$

Take the square root of 800.

$$x=\frac{-\left(-100\right)±20\sqrt{2}}{2}$$

The opposite of -100 is 100.

$$x=\frac{100±20\sqrt{2}}{2}$$

Now solve the equation x=\frac{100±20\sqrt{2}}{2} when ± is plus. Add 100 to 20\sqrt{2}.

$$x=\frac{20\sqrt{2}+100}{2}$$

Divide 100+20\sqrt{2} by 2.

$$x=10\sqrt{2}+50$$

Now solve the equation x=\frac{100±20\sqrt{2}}{2} when ± is minus. Subtract 20\sqrt{2} from 100.

$$x=\frac{100-20\sqrt{2}}{2}$$

Divide 100-20\sqrt{2} by 2.

$$x=50-10\sqrt{2}$$

The equation is now solved.

$$x=10\sqrt{2}+50 x=50-10\sqrt{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,100 since division by zero is not defined. Multiply both sides of the equation by 23x\left(x-100\right), the least common multiple of x,100-x,23.

$$23x-2300-23x=x\left(x-100\right)$$

Combine 23x and -23x to get 0.

$$-2300=x\left(x-100\right)$$

Use the distributive property to multiply x by x-100.

$$-2300=x^{2}-100x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-100x=-2300$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=-2300+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=-2300+2500$$

Add -2300 to 2500.

$$x^{2}-100x+2500=200$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=200$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{200}$$

Simplify.

$$x-50=10\sqrt{2} x-50=-10\sqrt{2}$$

Add 50 to both sides of the equation.

$$x=10\sqrt{2}+50 x=50-10\sqrt{2}$$