Solve 11-x=2sqrt{(x-3)(x-6)}

$$11 - x = 2 \sqrt { ( x - 3 ) ( x - 6 ) }$$

Solve for x

$$x = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

$$x=7$$

Solution Steps

Square both sides of the equation.

$$\left(11-x\right)^{2}=\left(2\sqrt{\left(x-3\right)\left(x-6\right)}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(11-x\right)^{2}.

$$121-22x+x^{2}=\left(2\sqrt{\left(x-3\right)\left(x-6\right)}\right)^{2}$$

Use the distributive property to multiply x-3 by x-6 and combine like terms.

$$121-22x+x^{2}=\left(2\sqrt{x^{2}-9x+18}\right)^{2}$$

Expand \left(2\sqrt{x^{2}-9x+18}\right)^{2}.

$$121-22x+x^{2}=2^{2}\left(\sqrt{x^{2}-9x+18}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$121-22x+x^{2}=4\left(\sqrt{x^{2}-9x+18}\right)^{2}$$

Calculate \sqrt{x^{2}-9x+18} to the power of 2 and get x^{2}-9x+18.

$$121-22x+x^{2}=4\left(x^{2}-9x+18\right)$$

Use the distributive property to multiply 4 by x^{2}-9x+18.

$$121-22x+x^{2}=4x^{2}-36x+72$$

Subtract 4x^{2} from both sides.

$$121-22x+x^{2}-4x^{2}=-36x+72$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$121-22x-3x^{2}=-36x+72$$

Add 36x to both sides.

$$121-22x-3x^{2}+36x=72$$

Combine -22x and 36x to get 14x.

$$121+14x-3x^{2}=72$$

Subtract 72 from both sides.

$$121+14x-3x^{2}-72=0$$

Subtract 72 from 121 to get 49.

$$49+14x-3x^{2}=0$$

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

$$-3x^{2}+14x+49=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx+49. To find a and b, set up a system to be solved.

$$a+b=14 ab=-3\times 49=-147$$

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 -147.

$$-1,147 -3,49 -7,21$$

Calculate the sum for each pair.

$$-1+147=146 -3+49=46 -7+21=14$$

The solution is the pair that gives sum 14.

$$a=21 b=-7$$

Rewrite -3x^{2}+14x+49 as \left(-3x^{2}+21x\right)+\left(-7x+49\right).

$$\left(-3x^{2}+21x\right)+\left(-7x+49\right)$$

Factor out 3x in the first and 7 in the second group.

$$3x\left(-x+7\right)+7\left(-x+7\right)$$

Factor out common term -x+7 by using distributive property.

$$\left(-x+7\right)\left(3x+7\right)$$

To find equation solutions, solve -x+7=0 and 3x+7=0.

$$x=7 x=-\frac{7}{3}$$

Substitute 7 for x in the equation 11-x=2\sqrt{\left(x-3\right)\left(x-6\right)}.

$$11-7=2\sqrt{\left(7-3\right)\left(7-6\right)}$$

Simplify. The value x=7 satisfies the equation.

$$4=4$$

Substitute -\frac{7}{3} for x in the equation 11-x=2\sqrt{\left(x-3\right)\left(x-6\right)}.

$$11-\left(-\frac{7}{3}\right)=2\sqrt{\left(-\frac{7}{3}-3\right)\left(-\frac{7}{3}-6\right)}$$

Simplify. The value x=-\frac{7}{3} satisfies the equation.

$$\frac{40}{3}=\frac{40}{3}$$

List all solutions of 11-x=2\sqrt{\left(x-6\right)\left(x-3\right)}.

$$x=7 x=-\frac{7}{3}$$