Solve sqrt{-x^2+9x-18}=3-x

$$\sqrt{ - { x }^{ 2 } +9x-18 } =3-x$$

Solve for x

$$x=3$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{-x^{2}+9x-18}\right)^{2}=\left(3-x\right)^{2}$$

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

$$-x^{2}+9x-18=\left(3-x\right)^{2}$$

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

$$-x^{2}+9x-18=9-6x+x^{2}$$

Subtract 9 from both sides.

$$-x^{2}+9x-18-9=-6x+x^{2}$$

Subtract 9 from -18 to get -27.

$$-x^{2}+9x-27=-6x+x^{2}$$

Add 6x to both sides.

$$-x^{2}+9x-27+6x=x^{2}$$

Combine 9x and 6x to get 15x.

$$-x^{2}+15x-27=x^{2}$$

Subtract x^{2} from both sides.

$$-x^{2}+15x-27-x^{2}=0$$

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

$$-2x^{2}+15x-27=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx-27. To find a and b, set up a system to be solved.

$$a+b=15 ab=-2\left(-27\right)=54$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 54.

$$1,54 2,27 3,18 6,9$$

Calculate the sum for each pair.

$$1+54=55 2+27=29 3+18=21 6+9=15$$

The solution is the pair that gives sum 15.

$$a=9 b=6$$

Rewrite -2x^{2}+15x-27 as \left(-2x^{2}+9x\right)+\left(6x-27\right).

$$\left(-2x^{2}+9x\right)+\left(6x-27\right)$$

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

$$-x\left(2x-9\right)+3\left(2x-9\right)$$

Factor out common term 2x-9 by using distributive property.

$$\left(2x-9\right)\left(-x+3\right)$$

To find equation solutions, solve 2x-9=0 and -x+3=0.

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

Substitute \frac{9}{2} for x in the equation \sqrt{-x^{2}+9x-18}=3-x.

$$\sqrt{-\left(\frac{9}{2}\right)^{2}+9\times \frac{9}{2}-18}=3-\frac{9}{2}$$

Simplify. The value x=\frac{9}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

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

Substitute 3 for x in the equation \sqrt{-x^{2}+9x-18}=3-x.

$$\sqrt{-3^{2}+9\times 3-18}=3-3$$

Simplify. The value x=3 satisfies the equation.

$$0=0$$

Equation \sqrt{-x^{2}+9x-18}=3-x has a unique solution.

$$x=3$$