Solve 4/x+9x=20

$$\frac{ 4 }{ x } +9x=20$$

Solve for x

$$x=\frac{2}{9}\approx 0.222222222$$

$$x=2$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$4+9xx=20x$$

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

$$4+9x^{2}=20x$$

Subtract 20x from both sides.

$$4+9x^{2}-20x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$9x^{2}-20x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-20 ab=9\times 4=36$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Calculate the sum for each pair.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

The solution is the pair that gives sum -20.

$$a=-18 b=-2$$

Rewrite 9x^{2}-20x+4 as \left(9x^{2}-18x\right)+\left(-2x+4\right).

$$\left(9x^{2}-18x\right)+\left(-2x+4\right)$$

Factor out 9x in the first and -2 in the second group.

$$9x\left(x-2\right)-2\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(9x-2\right)$$

To find equation solutions, solve x-2=0 and 9x-2=0.

$$x=2 x=\frac{2}{9}$$

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.

$$4+9xx=20x$$

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

$$4+9x^{2}=20x$$

Subtract 20x from both sides.

$$4+9x^{2}-20x=0$$

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.

$$9x^{2}-20x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -20 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 9\times 4}}{2\times 9}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 9\times 4}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-20\right)±\sqrt{400-36\times 4}}{2\times 9}$$

Multiply -36 times 4.

$$x=\frac{-\left(-20\right)±\sqrt{400-144}}{2\times 9}$$

Add 400 to -144.

$$x=\frac{-\left(-20\right)±\sqrt{256}}{2\times 9}$$

Take the square root of 256.

$$x=\frac{-\left(-20\right)±16}{2\times 9}$$

The opposite of -20 is 20.

$$x=\frac{20±16}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{20±16}{18}$$

Now solve the equation x=\frac{20±16}{18} when ± is plus. Add 20 to 16.

$$x=\frac{36}{18}$$

Divide 36 by 18.

$$x=2$$

Now solve the equation x=\frac{20±16}{18} when ± is minus. Subtract 16 from 20.

$$x=\frac{4}{18}$$

Reduce the fraction \frac{4}{18} to lowest terms by extracting and canceling out 2.

$$x=\frac{2}{9}$$

The equation is now solved.

$$x=2 x=\frac{2}{9}$$

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.

$$4+9xx=20x$$

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

$$4+9x^{2}=20x$$

Subtract 20x from both sides.

$$4+9x^{2}-20x=0$$

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

$$9x^{2}-20x=-4$$

Divide both sides by 9.

$$\frac{9x^{2}-20x}{9}=-\frac{4}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}-\frac{20}{9}x=-\frac{4}{9}$$

Divide -\frac{20}{9}, the coefficient of the x term, by 2 to get -\frac{10}{9}. Then add the square of -\frac{10}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{20}{9}x+\left(-\frac{10}{9}\right)^{2}=-\frac{4}{9}+\left(-\frac{10}{9}\right)^{2}$$

Square -\frac{10}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{20}{9}x+\frac{100}{81}=-\frac{4}{9}+\frac{100}{81}$$

Add -\frac{4}{9} to \frac{100}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{20}{9}x+\frac{100}{81}=\frac{64}{81}$$

Factor x^{2}-\frac{20}{9}x+\frac{100}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{10}{9}\right)^{2}=\frac{64}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{10}{9}\right)^{2}}=\sqrt{\frac{64}{81}}$$

Simplify.

$$x-\frac{10}{9}=\frac{8}{9} x-\frac{10}{9}=-\frac{8}{9}$$

Add \frac{10}{9} to both sides of the equation.

$$x=2 x=\frac{2}{9}$$