Solve 1/a+1/sqrt{a}-6=0

$$\frac { 1 } { a } + \frac { 1 } { \sqrt { a } } - 6 = 0$$

Solve for a

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

Solution Steps

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

$$1+a^{\frac{1}{2}}+a\left(-6\right)=0$$

Reorder the terms.

$$-6a+\sqrt{a}+1=0$$

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

$$-6a+\sqrt{a}=-1$$

Subtract -6a from both sides of the equation.

$$\sqrt{a}=-1+6a$$

Square both sides of the equation.

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

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

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

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

$$a=36a^{2}-12a+1$$

Subtract 36a^{2} from both sides.

$$a-36a^{2}=-12a+1$$

Add 12a to both sides.

$$a-36a^{2}+12a=1$$

Combine a and 12a to get 13a.

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

Subtract 1 from both sides.

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

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -36a^{2}+aa+ba-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 -36a^{2}+13a-1 as \left(-36a^{2}+9a\right)+\left(4a-1\right).

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

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

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

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

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

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

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

Substitute \frac{1}{4} for a in the equation \frac{1}{a}+\frac{1}{\sqrt{a}}-6=0.

$$\frac{1}{\frac{1}{4}}+\frac{1}{\sqrt{\frac{1}{4}}}-6=0$$

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

$$0=0$$

Substitute \frac{1}{9} for a in the equation \frac{1}{a}+\frac{1}{\sqrt{a}}-6=0.

$$\frac{1}{\frac{1}{9}}+\frac{1}{\sqrt{\frac{1}{9}}}-6=0$$

Simplify. The value a=\frac{1}{9} does not satisfy the equation.

$$6=0$$

Equation \sqrt{a}=6a-1 has a unique solution.

$$a=\frac{1}{4}$$