Solve 10=x+1/x

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

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.

$$10x=xx+1$$

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

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

Subtract x^{2} from both sides.

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

Subtract 1 from both sides.

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

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}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-10±\sqrt{100+4\left(-1\right)}}{2\left(-1\right)}$$

Multiply 4 times -1.

$$x=\frac{-10±\sqrt{100-4}}{2\left(-1\right)}$$

Add 100 to -4.

$$x=\frac{-10±\sqrt{96}}{2\left(-1\right)}$$

Take the square root of 96.

$$x=\frac{-10±4\sqrt{6}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-10±4\sqrt{6}}{-2}$$

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.

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

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

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

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

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

The equation is now solved.

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

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.

$$10x=xx+1$$

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

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

Subtract x^{2} from both sides.

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

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.

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

Divide both sides by -1.

$$\frac{-x^{2}+10x}{-1}=\frac{1}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{10}{-1}x=\frac{1}{-1}$$

Divide 10 by -1.

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

Divide 1 by -1.

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

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.

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