Solve x+1/x=10

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

Solve for x

$$x=2\sqrt{6}+5\approx 9.898979486$$

$$x=5-2\sqrt{6}\approx 0.101020514$$

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.

$$xx+1=10x$$

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

$$x^{2}+1=10x$$

Subtract 10x from both sides.

$$x^{2}+1-10x=0$$

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.

$$x^{2}-10x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4}}{2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4}}{2}$$

Add 100 to -4.

$$x=\frac{-\left(-10\right)±\sqrt{96}}{2}$$

Take the square root of 96.

$$x=\frac{-\left(-10\right)±4\sqrt{6}}{2}$$

The opposite of -10 is 10.

$$x=\frac{10±4\sqrt{6}}{2}$$

Now solve the equation x=\frac{10±4\sqrt{6}}{2} when ± is plus. Add 10 to 4\sqrt{6}.

$$x=\frac{4\sqrt{6}+10}{2}$$

Divide 10+4\sqrt{6} by 2.

$$x=2\sqrt{6}+5$$

Now solve the equation x=\frac{10±4\sqrt{6}}{2} when ± is minus. Subtract 4\sqrt{6} from 10.

$$x=\frac{10-4\sqrt{6}}{2}$$

Divide 10-4\sqrt{6} by 2.

$$x=5-2\sqrt{6}$$

The equation is now solved.

$$x=2\sqrt{6}+5 x=5-2\sqrt{6}$$

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.

$$xx+1=10x$$

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

$$x^{2}+1=10x$$

Subtract 10x from both sides.

$$x^{2}+1-10x=0$$

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

$$x^{2}-10x=-1$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-1+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-1+25$$

Add -1 to 25.

$$x^{2}-10x+25=24$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=24$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{24}$$

Simplify.

$$x-5=2\sqrt{6} x-5=-2\sqrt{6}$$

Add 5 to both sides of the equation.

$$x=2\sqrt{6}+5 x=5-2\sqrt{6}$$