Solve 6x=1-sqrt{x}

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

Solve for x

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

Solution Steps

Subtract 1 from both sides of the equation.

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

Square both sides of the equation.

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

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

$$36x^{2}-12x+1=\left(-\sqrt{x}\right)^{2}$$

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

$$36x^{2}-12x+1=\left(-1\right)^{2}\left(\sqrt{x}\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$36x^{2}-12x+1=1\left(\sqrt{x}\right)^{2}$$

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

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

Reorder the terms.

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

Subtract x from both sides.

$$36x^{2}-12x+1-x=0$$

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

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

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.

$$a+b=-13 ab=36\times 1=36$$

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 -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 the first and -1 in the second group.

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

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 6x=1-\sqrt{x}.

$$6\times \frac{1}{4}=1-\sqrt{\frac{1}{4}}$$

Simplify. The value x=\frac{1}{4} does not satisfy the equation.

$$\frac{3}{2}=\frac{1}{2}$$

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

$$6\times \frac{1}{9}=1-\sqrt{\frac{1}{9}}$$

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

$$\frac{2}{3}=\frac{2}{3}$$

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

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