Solve 2(x-3)^2=-x(3-x)

$$2 ( x - 3 ) ^ { 2 } = - x ( 3 - x )$$

Solve for x

$$x=3$$

$$x=6$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$2\left(x^{2}-6x+9\right)=\left(-x\right)\left(3-x\right)$$

Use the distributive property to multiply 2 by x^{2}-6x+9.

$$2x^{2}-12x+18=\left(-x\right)\left(3-x\right)$$

Use the distributive property to multiply -x by 3-x.

$$2x^{2}-12x+18=3\left(-x\right)-\left(-x\right)x$$

Multiply -1 and -1 to get 1.

$$2x^{2}-12x+18=3\left(-x\right)+xx$$

Multiply x and x to get x^{2}.

$$2x^{2}-12x+18=3\left(-x\right)+x^{2}$$

Subtract 3\left(-x\right) from both sides.

$$2x^{2}-12x+18-3\left(-x\right)=x^{2}$$

Subtract x^{2} from both sides.

$$2x^{2}-12x+18-3\left(-x\right)-x^{2}=0$$

Multiply -1 and 3 to get -3.

$$2x^{2}-12x+18-3\left(-1\right)x-x^{2}=0$$

Multiply -3 and -1 to get 3.

$$2x^{2}-12x+18+3x-x^{2}=0$$

Combine -12x and 3x to get -9x.

$$2x^{2}-9x+18-x^{2}=0$$

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

$$x^{2}-9x+18=0$$

To solve the equation, factor x^{2}-9x+18 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-9 ab=18$$

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 18.

$$-1,-18 -2,-9 -3,-6$$

Calculate the sum for each pair.

$$-1-18=-19 -2-9=-11 -3-6=-9$$

The solution is the pair that gives sum -9.

$$a=-6 b=-3$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-6\right)\left(x-3\right)$$

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

$$x=6 x=3$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$2\left(x^{2}-6x+9\right)=\left(-x\right)\left(3-x\right)$$

Use the distributive property to multiply 2 by x^{2}-6x+9.

$$2x^{2}-12x+18=\left(-x\right)\left(3-x\right)$$

Use the distributive property to multiply -x by 3-x.

$$2x^{2}-12x+18=3\left(-x\right)-\left(-x\right)x$$

Multiply -1 and -1 to get 1.

$$2x^{2}-12x+18=3\left(-x\right)+xx$$

Multiply x and x to get x^{2}.

$$2x^{2}-12x+18=3\left(-x\right)+x^{2}$$

Subtract 3\left(-x\right) from both sides.

$$2x^{2}-12x+18-3\left(-x\right)=x^{2}$$

Subtract x^{2} from both sides.

$$2x^{2}-12x+18-3\left(-x\right)-x^{2}=0$$

Multiply -1 and 3 to get -3.

$$2x^{2}-12x+18-3\left(-1\right)x-x^{2}=0$$

Multiply -3 and -1 to get 3.

$$2x^{2}-12x+18+3x-x^{2}=0$$

Combine -12x and 3x to get -9x.

$$2x^{2}-9x+18-x^{2}=0$$

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

$$x^{2}-9x+18=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+18. To find a and b, set up a system to be solved.

$$a+b=-9 ab=1\times 18=18$$

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 18.

$$-1,-18 -2,-9 -3,-6$$

Calculate the sum for each pair.

$$-1-18=-19 -2-9=-11 -3-6=-9$$

The solution is the pair that gives sum -9.

$$a=-6 b=-3$$

Rewrite x^{2}-9x+18 as \left(x^{2}-6x\right)+\left(-3x+18\right).

$$\left(x^{2}-6x\right)+\left(-3x+18\right)$$

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

$$x\left(x-6\right)-3\left(x-6\right)$$

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

$$\left(x-6\right)\left(x-3\right)$$

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

$$x=6 x=3$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$2\left(x^{2}-6x+9\right)=\left(-x\right)\left(3-x\right)$$

Use the distributive property to multiply 2 by x^{2}-6x+9.

$$2x^{2}-12x+18=\left(-x\right)\left(3-x\right)$$

Use the distributive property to multiply -x by 3-x.

$$2x^{2}-12x+18=3\left(-x\right)-\left(-x\right)x$$

Multiply -1 and -1 to get 1.

$$2x^{2}-12x+18=3\left(-x\right)+xx$$

Multiply x and x to get x^{2}.

$$2x^{2}-12x+18=3\left(-x\right)+x^{2}$$

Subtract 3\left(-x\right) from both sides.

$$2x^{2}-12x+18-3\left(-x\right)=x^{2}$$

Subtract x^{2} from both sides.

$$2x^{2}-12x+18-3\left(-x\right)-x^{2}=0$$

Multiply -1 and 3 to get -3.

$$2x^{2}-12x+18-3\left(-1\right)x-x^{2}=0$$

Multiply -3 and -1 to get 3.

$$2x^{2}-12x+18+3x-x^{2}=0$$

Combine -12x and 3x to get -9x.

$$2x^{2}-9x+18-x^{2}=0$$

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

$$x^{2}-9x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 18}}{2}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 18}}{2}$$

Multiply -4 times 18.

$$x=\frac{-\left(-9\right)±\sqrt{81-72}}{2}$$

Add 81 to -72.

$$x=\frac{-\left(-9\right)±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-\left(-9\right)±3}{2}$$

The opposite of -9 is 9.

$$x=\frac{9±3}{2}$$

Now solve the equation x=\frac{9±3}{2} when ± is plus. Add 9 to 3.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Now solve the equation x=\frac{9±3}{2} when ± is minus. Subtract 3 from 9.

$$x=\frac{6}{2}$$

Divide 6 by 2.

$$x=3$$

The equation is now solved.

$$x=6 x=3$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$2\left(x^{2}-6x+9\right)=\left(-x\right)\left(3-x\right)$$

Use the distributive property to multiply 2 by x^{2}-6x+9.

$$2x^{2}-12x+18=\left(-x\right)\left(3-x\right)$$

Use the distributive property to multiply -x by 3-x.

$$2x^{2}-12x+18=3\left(-x\right)-\left(-x\right)x$$

Multiply -1 and -1 to get 1.

$$2x^{2}-12x+18=3\left(-x\right)+xx$$

Multiply x and x to get x^{2}.

$$2x^{2}-12x+18=3\left(-x\right)+x^{2}$$

Subtract 3\left(-x\right) from both sides.

$$2x^{2}-12x+18-3\left(-x\right)=x^{2}$$

Subtract x^{2} from both sides.

$$2x^{2}-12x+18-3\left(-x\right)-x^{2}=0$$

Multiply -1 and 3 to get -3.

$$2x^{2}-12x+18-3\left(-1\right)x-x^{2}=0$$

Multiply -3 and -1 to get 3.

$$2x^{2}-12x+18+3x-x^{2}=0$$

Combine -12x and 3x to get -9x.

$$2x^{2}-9x+18-x^{2}=0$$

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

$$x^{2}-9x+18=0$$

Subtract 18 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-9x=-18$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-18+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=-18+\frac{81}{4}$$

Add -18 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{9}{4}$$

Factor x^{2}-9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{3}{2} x-\frac{9}{2}=-\frac{3}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=6 x=3$$