Solve x+1/x=100

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

Solve for x

$$x=7\sqrt{51}+50\approx 99.989999$$

$$x=50-7\sqrt{51}\approx 0.010001$$

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 x.

$$xx+1=100x$$

Multiply x and x to get x^{2}.

$$x^{2}+1=100x$$

Subtract 100x from both sides.

$$x^{2}+1-100x=0$$

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.

$$x^{2}-100x+1=0$$

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

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

Square -100.

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

Add 10000 to -4.

$$x=\frac{-\left(-100\right)±\sqrt{9996}}{2}$$

Take the square root of 9996.

$$x=\frac{-\left(-100\right)±14\sqrt{51}}{2}$$

The opposite of -100 is 100.

$$x=\frac{100±14\sqrt{51}}{2}$$

Now solve the equation x=\frac{100±14\sqrt{51}}{2} when ± is plus. Add 100 to 14\sqrt{51}.

$$x=\frac{14\sqrt{51}+100}{2}$$

Divide 100+14\sqrt{51} by 2.

$$x=7\sqrt{51}+50$$

Now solve the equation x=\frac{100±14\sqrt{51}}{2} when ± is minus. Subtract 14\sqrt{51} from 100.

$$x=\frac{100-14\sqrt{51}}{2}$$

Divide 100-14\sqrt{51} by 2.

$$x=50-7\sqrt{51}$$

The equation is now solved.

$$x=7\sqrt{51}+50 x=50-7\sqrt{51}$$

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 x.

$$xx+1=100x$$

Multiply x and x to get x^{2}.

$$x^{2}+1=100x$$

Subtract 100x from both sides.

$$x^{2}+1-100x=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-100x=-1$$

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.

$$x^{2}-100x+\left(-50\right)^{2}=-1+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=-1+2500$$

Add -1 to 2500.

$$x^{2}-100x+2500=2499$$

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}.

$$\left(x-50\right)^{2}=2499$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{2499}$$

Simplify.

$$x-50=7\sqrt{51} x-50=-7\sqrt{51}$$

Add 50 to both sides of the equation.

$$x=7\sqrt{51}+50 x=50-7\sqrt{51}$$