Solve 6x=sqrt{6-6x}

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

Solve for x

$$x=\frac{1}{3}\approx 0.333333333$$

Solution Steps

Square both sides of the equation.

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

Expand \left(6x\right)^{2}.

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

Calculate 6 to the power of 2 and get 36.

$$36x^{2}=\left(\sqrt{6-6x}\right)^{2}$$

Calculate \sqrt{6-6x} to the power of 2 and get 6-6x.

$$36x^{2}=6-6x$$

Subtract 6 from both sides.

$$36x^{2}-6=-6x$$

Add 6x to both sides.

$$36x^{2}-6+6x=0$$

Divide both sides by 6.

$$6x^{2}-1+x=0$$

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

$$6x^{2}+x-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=1 ab=6\left(-1\right)=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 1.

$$a=-2 b=3$$

Rewrite 6x^{2}+x-1 as \left(6x^{2}-2x\right)+\left(3x-1\right).

$$\left(6x^{2}-2x\right)+\left(3x-1\right)$$

Factor out 2x in 6x^{2}-2x.

$$2x\left(3x-1\right)+3x-1$$

Factor out common term 3x-1 by using distributive property.

$$\left(3x-1\right)\left(2x+1\right)$$

To find equation solutions, solve 3x-1=0 and 2x+1=0.

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

Substitute \frac{1}{3} for x in the equation 6x=\sqrt{6-6x}.

$$6\times \frac{1}{3}=\sqrt{6-6\times \frac{1}{3}}$$

Simplify. The value x=\frac{1}{3} satisfies the equation.

$$2=2$$

Substitute -\frac{1}{2} for x in the equation 6x=\sqrt{6-6x}.

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

Simplify. The value x=-\frac{1}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-3=3$$

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

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