Solve (6sqrt{3})^2=(6+x/2)2x

$$( 6 \sqrt { 3 } ) ^ { 2 } = ( 6 + \frac { x } { 2 } ) 2 x$$

Solve for x

$$x=-18$$

$$x=6$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 2.

$$2\times \left(6\sqrt{3}\right)^{2}=2\left(6+\frac{x}{2}\right)\times 2x$$

Expand \left(6\sqrt{3}\right)^{2}.

$$2\times 6^{2}\left(\sqrt{3}\right)^{2}=2\left(6+\frac{x}{2}\right)\times 2x$$

Calculate 6 to the power of 2 and get 36.

$$2\times 36\left(\sqrt{3}\right)^{2}=2\left(6+\frac{x}{2}\right)\times 2x$$

The square of \sqrt{3} is 3.

$$2\times 36\times 3=2\left(6+\frac{x}{2}\right)\times 2x$$

Multiply 36 and 3 to get 108.

$$2\times 108=2\left(6+\frac{x}{2}\right)\times 2x$$

Multiply 2 and 108 to get 216.

$$216=2\left(6+\frac{x}{2}\right)\times 2x$$

Multiply 2 and 2 to get 4.

$$216=4\left(6+\frac{x}{2}\right)x$$

Use the distributive property to multiply 4 by 6+\frac{x}{2}.

$$216=\left(24+4\times \frac{x}{2}\right)x$$

Cancel out 2, the greatest common factor in 4 and 2.

$$216=\left(24+2x\right)x$$

Use the distributive property to multiply 24+2x by x.

$$216=24x+2x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$24x+2x^{2}=216$$

Subtract 216 from both sides.

$$24x+2x^{2}-216=0$$

Divide both sides by 2.

$$12x+x^{2}-108=0$$

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

$$x^{2}+12x-108=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-108. To find a and b, set up a system to be solved.

$$a+b=12 ab=1\left(-108\right)=-108$$

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 -108.

$$-1,108 -2,54 -3,36 -4,27 -6,18 -9,12$$

Calculate the sum for each pair.

$$-1+108=107 -2+54=52 -3+36=33 -4+27=23 -6+18=12 -9+12=3$$

The solution is the pair that gives sum 12.

$$a=-6 b=18$$

Rewrite x^{2}+12x-108 as \left(x^{2}-6x\right)+\left(18x-108\right).

$$\left(x^{2}-6x\right)+\left(18x-108\right)$$

Factor out x in the first and 18 in the second group.

$$x\left(x-6\right)+18\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(x+18\right)$$

To find equation solutions, solve x-6=0 and x+18=0.

$$x=6 x=-18$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2\times \left(6\sqrt{3}\right)^{2}=2\left(6+\frac{x}{2}\right)\times 2x$$

Expand \left(6\sqrt{3}\right)^{2}.

$$2\times 6^{2}\left(\sqrt{3}\right)^{2}=2\left(6+\frac{x}{2}\right)\times 2x$$

Calculate 6 to the power of 2 and get 36.

$$2\times 36\left(\sqrt{3}\right)^{2}=2\left(6+\frac{x}{2}\right)\times 2x$$

The square of \sqrt{3} is 3.

$$2\times 36\times 3=2\left(6+\frac{x}{2}\right)\times 2x$$

Multiply 36 and 3 to get 108.

$$2\times 108=2\left(6+\frac{x}{2}\right)\times 2x$$

Multiply 2 and 108 to get 216.

$$216=2\left(6+\frac{x}{2}\right)\times 2x$$

Multiply 2 and 2 to get 4.

$$216=4\left(6+\frac{x}{2}\right)x$$

Use the distributive property to multiply 4 by 6+\frac{x}{2}.

$$216=\left(24+4\times \frac{x}{2}\right)x$$

Cancel out 2, the greatest common factor in 4 and 2.

$$216=\left(24+2x\right)x$$

Use the distributive property to multiply 24+2x by x.

$$216=24x+2x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$24x+2x^{2}=216$$

Subtract 216 from both sides.

$$24x+2x^{2}-216=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$2x^{2}+24x-216=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 24 for b, and -216 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 2\left(-216\right)}}{2\times 2}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 2\left(-216\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-24±\sqrt{576-8\left(-216\right)}}{2\times 2}$$

Multiply -8 times -216.

$$x=\frac{-24±\sqrt{576+1728}}{2\times 2}$$

Add 576 to 1728.

$$x=\frac{-24±\sqrt{2304}}{2\times 2}$$

Take the square root of 2304.

$$x=\frac{-24±48}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-24±48}{4}$$

Now solve the equation x=\frac{-24±48}{4} when ± is plus. Add -24 to 48.

$$x=\frac{24}{4}$$

Divide 24 by 4.

$$x=6$$

Now solve the equation x=\frac{-24±48}{4} when ± is minus. Subtract 48 from -24.

$$x=-\frac{72}{4}$$

Divide -72 by 4.

$$x=-18$$

The equation is now solved.

$$x=6 x=-18$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2\times \left(6\sqrt{3}\right)^{2}=2\left(6+\frac{x}{2}\right)\times 2x$$

Expand \left(6\sqrt{3}\right)^{2}.

$$2\times 6^{2}\left(\sqrt{3}\right)^{2}=2\left(6+\frac{x}{2}\right)\times 2x$$

Calculate 6 to the power of 2 and get 36.

$$2\times 36\left(\sqrt{3}\right)^{2}=2\left(6+\frac{x}{2}\right)\times 2x$$

The square of \sqrt{3} is 3.

$$2\times 36\times 3=2\left(6+\frac{x}{2}\right)\times 2x$$

Multiply 36 and 3 to get 108.

$$2\times 108=2\left(6+\frac{x}{2}\right)\times 2x$$

Multiply 2 and 108 to get 216.

$$216=2\left(6+\frac{x}{2}\right)\times 2x$$

Multiply 2 and 2 to get 4.

$$216=4\left(6+\frac{x}{2}\right)x$$

Use the distributive property to multiply 4 by 6+\frac{x}{2}.

$$216=\left(24+4\times \frac{x}{2}\right)x$$

Cancel out 2, the greatest common factor in 4 and 2.

$$216=\left(24+2x\right)x$$

Use the distributive property to multiply 24+2x by x.

$$216=24x+2x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$24x+2x^{2}=216$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$2x^{2}+24x=216$$

Divide both sides by 2.

$$\frac{2x^{2}+24x}{2}=\frac{216}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{24}{2}x=\frac{216}{2}$$

Divide 24 by 2.

$$x^{2}+12x=\frac{216}{2}$$

Divide 216 by 2.

$$x^{2}+12x=108$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=108+6^{2}$$

Square 6.

$$x^{2}+12x+36=108+36$$

Add 108 to 36.

$$x^{2}+12x+36=144$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=144$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{144}$$

Simplify.

$$x+6=12 x+6=-12$$

Subtract 6 from both sides of the equation.

$$x=6 x=-18$$