Solve left(6-xright)^2=(3-x)(12+2x)

$${ \left(6-x \right) }^{ 2 } =(3-x)(12+2x)$$

Solve for x

$$x=2$$

$$x=0$$

Steps Using Factoring

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

$$36-12x+x^{2}=\left(3-x\right)\left(12+2x\right)$$

Use the distributive property to multiply 3-x by 12+2x and combine like terms.

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

Subtract 36 from both sides.

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

Subtract 36 from 36 to get 0.

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

Add 6x to both sides.

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

Combine -12x and 6x to get -6x.

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

Add 2x^{2} to both sides.

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

Combine x^{2} and 2x^{2} to get 3x^{2}.

$$-6x+3x^{2}=0$$

Factor out x.

$$x\left(-6+3x\right)=0$$

To find equation solutions, solve x=0 and -6+3x=0.

$$x=0 x=2$$

Steps Using the Quadratic Formula

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

$$36-12x+x^{2}=\left(3-x\right)\left(12+2x\right)$$

Use the distributive property to multiply 3-x by 12+2x and combine like terms.

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

Subtract 36 from both sides.

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

Subtract 36 from 36 to get 0.

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

Add 6x to both sides.

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

Combine -12x and 6x to get -6x.

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

Add 2x^{2} to both sides.

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

Combine x^{2} and 2x^{2} to get 3x^{2}.

$$-6x+3x^{2}=0$$

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.

$$3x^{2}-6x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -6 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Take the square root of \left(-6\right)^{2}.

$$x=\frac{-\left(-6\right)±6}{2\times 3}$$

The opposite of -6 is 6.

$$x=\frac{6±6}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{6±6}{6}$$

Now solve the equation x=\frac{6±6}{6} when ± is plus. Add 6 to 6.

$$x=\frac{12}{6}$$

Divide 12 by 6.

$$x=2$$

Now solve the equation x=\frac{6±6}{6} when ± is minus. Subtract 6 from 6.

$$x=\frac{0}{6}$$

Divide 0 by 6.

$$x=0$$

The equation is now solved.

$$x=2 x=0$$

Steps for Completing the Square

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

$$36-12x+x^{2}=\left(3-x\right)\left(12+2x\right)$$

Use the distributive property to multiply 3-x by 12+2x and combine like terms.

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

Add 6x to both sides.

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

Combine -12x and 6x to get -6x.

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

Add 2x^{2} to both sides.

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

Combine x^{2} and 2x^{2} to get 3x^{2}.

$$36-6x+3x^{2}=36$$

Subtract 36 from both sides.

$$-6x+3x^{2}=36-36$$

Subtract 36 from 36 to get 0.

$$-6x+3x^{2}=0$$

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.

$$3x^{2}-6x=0$$

Divide both sides by 3.

$$\frac{3x^{2}-6x}{3}=\frac{0}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{6}{3}\right)x=\frac{0}{3}$$

Divide -6 by 3.

$$x^{2}-2x=\frac{0}{3}$$

Divide 0 by 3.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-1=1 x-1=-1$$

Add 1 to both sides of the equation.

$$x=2 x=0$$