Solve 1/x+6=5/sqrt{x}

$$\frac { 1 } { x } + 6 = \frac { 5 } { \sqrt { x } }$$

Solve for x

$$x=\frac{1}{9}\approx 0.111111111$$

$$x=\frac{1}{4}=0.25$$

Solution Steps

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

$$1+x\times 6=x^{\frac{1}{2}}\times 5$$

Subtract x^{\frac{1}{2}}\times 5 from both sides.

$$1+x\times 6-x^{\frac{1}{2}}\times 5=0$$

Reorder the terms.

$$6x+1-5\sqrt{x}=0$$

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

$$6x-5\sqrt{x}=-1$$

Subtract 6x from both sides of the equation.

$$-5\sqrt{x}=-1-6x$$

Square both sides of the equation.

$$\left(-5\sqrt{x}\right)^{2}=\left(-1-6x\right)^{2}$$

Expand \left(-5\sqrt{x}\right)^{2}.

$$\left(-5\right)^{2}\left(\sqrt{x}\right)^{2}=\left(-1-6x\right)^{2}$$

Calculate -5 to the power of 2 and get 25.

$$25\left(\sqrt{x}\right)^{2}=\left(-1-6x\right)^{2}$$

Calculate \sqrt{x} to the power of 2 and get x.

$$25x=\left(-1-6x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-1-6x\right)^{2}.

$$25x=1+12x+36x^{2}$$

Subtract 12x from both sides.

$$25x-12x=1+36x^{2}$$

Combine 25x and -12x to get 13x.

$$13x=1+36x^{2}$$

Subtract 36x^{2} from both sides.

$$13x-36x^{2}=1$$

Subtract 1 from both sides.

$$13x-36x^{2}-1=0$$

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

$$-36x^{2}+13x-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -36x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=13 ab=-36\left(-1\right)=36$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. 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 13.

$$a=9 b=4$$

Rewrite -36x^{2}+13x-1 as \left(-36x^{2}+9x\right)+\left(4x-1\right).

$$\left(-36x^{2}+9x\right)+\left(4x-1\right)$$

Factor out -9x in -36x^{2}+9x.

$$-9x\left(4x-1\right)+4x-1$$

Factor out common term 4x-1 by using distributive property.

$$\left(4x-1\right)\left(-9x+1\right)$$

To find equation solutions, solve 4x-1=0 and -9x+1=0.

$$x=\frac{1}{4} x=\frac{1}{9}$$

Substitute \frac{1}{4} for x in the equation \frac{1}{x}+6=\frac{5}{\sqrt{x}}.

$$\frac{1}{\frac{1}{4}}+6=\frac{5}{\sqrt{\frac{1}{4}}}$$

Simplify. The value x=\frac{1}{4} satisfies the equation.

$$10=10$$

Substitute \frac{1}{9} for x in the equation \frac{1}{x}+6=\frac{5}{\sqrt{x}}.

$$\frac{1}{\frac{1}{9}}+6=\frac{5}{\sqrt{\frac{1}{9}}}$$

Simplify. The value x=\frac{1}{9} satisfies the equation.

$$15=15$$

List all solutions of -5\sqrt{x}=-6x-1.

$$x=\frac{1}{4} x=\frac{1}{9}$$