Solve 3/x+x=101

$$\frac { 3 } { x } + x = 101$$

Solve for x

$$x = \frac{\sqrt{10189} + 101}{2} \approx 100.970288289$$

$$x=\frac{101-\sqrt{10189}}{2}\approx 0.029711711$$

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.

$$3+xx=101x$$

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

$$3+x^{2}=101x$$

Subtract 101x from both sides.

$$3+x^{2}-101x=0$$

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.

$$x^{2}-101x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -101 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-101\right)±\sqrt{\left(-101\right)^{2}-4\times 3}}{2}$$

Square -101.

$$x=\frac{-\left(-101\right)±\sqrt{10201-4\times 3}}{2}$$

Multiply -4 times 3.

$$x=\frac{-\left(-101\right)±\sqrt{10201-12}}{2}$$

Add 10201 to -12.

$$x=\frac{-\left(-101\right)±\sqrt{10189}}{2}$$

The opposite of -101 is 101.

$$x=\frac{101±\sqrt{10189}}{2}$$

Now solve the equation x=\frac{101±\sqrt{10189}}{2} when ± is plus. Add 101 to \sqrt{10189}.

$$x=\frac{\sqrt{10189}+101}{2}$$

Now solve the equation x=\frac{101±\sqrt{10189}}{2} when ± is minus. Subtract \sqrt{10189} from 101.

$$x=\frac{101-\sqrt{10189}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{10189}+101}{2} x=\frac{101-\sqrt{10189}}{2}$$

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.

$$3+xx=101x$$

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

$$3+x^{2}=101x$$

Subtract 101x from both sides.

$$3+x^{2}-101x=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-101x=-3$$

Divide -101, the coefficient of the x term, by 2 to get -\frac{101}{2}. Then add the square of -\frac{101}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-101x+\left(-\frac{101}{2}\right)^{2}=-3+\left(-\frac{101}{2}\right)^{2}$$

Square -\frac{101}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-101x+\frac{10201}{4}=-3+\frac{10201}{4}$$

Add -3 to \frac{10201}{4}.

$$x^{2}-101x+\frac{10201}{4}=\frac{10189}{4}$$

Factor x^{2}-101x+\frac{10201}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{101}{2}\right)^{2}=\frac{10189}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{101}{2}\right)^{2}}=\sqrt{\frac{10189}{4}}$$

Simplify.

$$x-\frac{101}{2}=\frac{\sqrt{10189}}{2} x-\frac{101}{2}=-\frac{\sqrt{10189}}{2}$$

Add \frac{101}{2} to both sides of the equation.

$$x=\frac{\sqrt{10189}+101}{2} x=\frac{101-\sqrt{10189}}{2}$$