Solve x+4/x=29

$$x+ \frac{ 4 }{ x } =29$$

Solve for x

$$x = \frac{5 \sqrt{33} + 29}{2} \approx 28.861406616$$

$$x=\frac{29-5\sqrt{33}}{2}\approx 0.138593384$$

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.

$$xx+4=29x$$

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

$$x^{2}+4=29x$$

Subtract 29x from both sides.

$$x^{2}+4-29x=0$$

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.

$$x^{2}-29x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -29 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\times 4}}{2}$$

Square -29.

$$x=\frac{-\left(-29\right)±\sqrt{841-4\times 4}}{2}$$

Multiply -4 times 4.

$$x=\frac{-\left(-29\right)±\sqrt{841-16}}{2}$$

Add 841 to -16.

$$x=\frac{-\left(-29\right)±\sqrt{825}}{2}$$

Take the square root of 825.

$$x=\frac{-\left(-29\right)±5\sqrt{33}}{2}$$

The opposite of -29 is 29.

$$x=\frac{29±5\sqrt{33}}{2}$$

Now solve the equation x=\frac{29±5\sqrt{33}}{2} when ± is plus. Add 29 to 5\sqrt{33}.

$$x=\frac{5\sqrt{33}+29}{2}$$

Now solve the equation x=\frac{29±5\sqrt{33}}{2} when ± is minus. Subtract 5\sqrt{33} from 29.

$$x=\frac{29-5\sqrt{33}}{2}$$

The equation is now solved.

$$x=\frac{5\sqrt{33}+29}{2} x=\frac{29-5\sqrt{33}}{2}$$

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.

$$xx+4=29x$$

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

$$x^{2}+4=29x$$

Subtract 29x from both sides.

$$x^{2}+4-29x=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-29x=-4$$

Divide -29, the coefficient of the x term, by 2 to get -\frac{29}{2}. Then add the square of -\frac{29}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-29x+\left(-\frac{29}{2}\right)^{2}=-4+\left(-\frac{29}{2}\right)^{2}$$

Square -\frac{29}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-29x+\frac{841}{4}=-4+\frac{841}{4}$$

Add -4 to \frac{841}{4}.

$$x^{2}-29x+\frac{841}{4}=\frac{825}{4}$$

Factor x^{2}-29x+\frac{841}{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}.

$$\left(x-\frac{29}{2}\right)^{2}=\frac{825}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{29}{2}\right)^{2}}=\sqrt{\frac{825}{4}}$$

Simplify.

$$x-\frac{29}{2}=\frac{5\sqrt{33}}{2} x-\frac{29}{2}=-\frac{5\sqrt{33}}{2}$$

Add \frac{29}{2} to both sides of the equation.

$$x=\frac{5\sqrt{33}+29}{2} x=\frac{29-5\sqrt{33}}{2}$$