Solve 1/x-4/sqrt{x}+3=0

$$\frac{ 1 }{ x } - \frac{ 4 }{ \sqrt{ x } } +3 = 0$$

Solve for x

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

$$x=1$$

Solution Steps

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

$$1-x^{\frac{1}{2}}\times 4+x\times 3=0$$

Reorder the terms.

$$3x-4\sqrt{x}+1=0$$

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

$$3x-4\sqrt{x}=-1$$

Subtract 3x from both sides of the equation.

$$-4\sqrt{x}=-1-3x$$

Square both sides of the equation.

$$\left(-4\sqrt{x}\right)^{2}=\left(-1-3x\right)^{2}$$

Expand \left(-4\sqrt{x}\right)^{2}.

$$\left(-4\right)^{2}\left(\sqrt{x}\right)^{2}=\left(-1-3x\right)^{2}$$

Calculate -4 to the power of 2 and get 16.

$$16\left(\sqrt{x}\right)^{2}=\left(-1-3x\right)^{2}$$

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

$$16x=\left(-1-3x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-1-3x\right)^{2}.

$$16x=1+6x+9x^{2}$$

Subtract 6x from both sides.

$$16x-6x=1+9x^{2}$$

Combine 16x and -6x to get 10x.

$$10x=1+9x^{2}$$

Subtract 9x^{2} from both sides.

$$10x-9x^{2}=1$$

Subtract 1 from both sides.

$$10x-9x^{2}-1=0$$

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

$$-9x^{2}+10x-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=10 ab=-9\left(-1\right)=9$$

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 9.

$$1,9 3,3$$

Calculate the sum for each pair.

$$1+9=10 3+3=6$$

The solution is the pair that gives sum 10.

$$a=9 b=1$$

Rewrite -9x^{2}+10x-1 as \left(-9x^{2}+9x\right)+\left(x-1\right).

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

Factor out 9x in the first and -1 in the second group.

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

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

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

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

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

Substitute 1 for x in the equation \frac{1}{x}-\frac{4}{\sqrt{x}}+3=0.

$$\frac{1}{1}-\frac{4}{\sqrt{1}}+3=0$$

Simplify. The value x=1 satisfies the equation.

$$0=0$$

Substitute \frac{1}{9} for x in the equation \frac{1}{x}-\frac{4}{\sqrt{x}}+3=0.

$$\frac{1}{\frac{1}{9}}-\frac{4}{\sqrt{\frac{1}{9}}}+3=0$$

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

$$0=0$$

List all solutions of -4\sqrt{x}=-3x-1.

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