Solve 6x=sqrt{12-6x}

$$6 x = \sqrt { 12 - 6 x }$$

Solve for x

$$x=\frac{1}{2}=0.5$$

Solution Steps

Square both sides of the equation.

$$\left(6x\right)^{2}=\left(\sqrt{12-6x}\right)^{2}$$

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

$$6^{2}x^{2}=\left(\sqrt{12-6x}\right)^{2}$$

Calculate 6 to the power of 2 and get 36.

$$36x^{2}=\left(\sqrt{12-6x}\right)^{2}$$

Calculate \sqrt{12-6x} to the power of 2 and get 12-6x.

$$36x^{2}=12-6x$$

Subtract 12 from both sides.

$$36x^{2}-12=-6x$$

Add 6x to both sides.

$$36x^{2}-12+6x=0$$

Divide both sides by 6.

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

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

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

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

$$a+b=1 ab=6\left(-2\right)=-12$$

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 -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 1.

$$a=-3 b=4$$

Rewrite 6x^{2}+x-2 as \left(6x^{2}-3x\right)+\left(4x-2\right).

$$\left(6x^{2}-3x\right)+\left(4x-2\right)$$

Factor out 3x in the first and 2 in the second group.

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

Factor out common term 2x-1 by using distributive property.

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

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

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

Substitute \frac{1}{2} for x in the equation 6x=\sqrt{12-6x}.

$$6\times \frac{1}{2}=\sqrt{12-6\times \frac{1}{2}}$$

Simplify. The value x=\frac{1}{2} satisfies the equation.

$$3=3$$

Substitute -\frac{2}{3} for x in the equation 6x=\sqrt{12-6x}.

$$6\left(-\frac{2}{3}\right)=\sqrt{12-6\left(-\frac{2}{3}\right)}$$

Simplify. The value x=-\frac{2}{3} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-4=4$$

Equation 6x=\sqrt{12-6x} has a unique solution.

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