Solve 3-sqrt{x-2}=5-x

$$3 - \sqrt { x - 2 } = 5 - x$$

Solve for x

$$x=3$$

$$x=2$$

Solution Steps

Subtract 3 from both sides of the equation.

$$-\sqrt{x-2}=5-x-3$$

Subtract 3 from 5 to get 2.

$$-\sqrt{x-2}=2-x$$

Square both sides of the equation.

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

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

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

Calculate -1 to the power of 2 and get 1.

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

Calculate \sqrt{x-2} to the power of 2 and get x-2.

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

Use the distributive property to multiply 1 by x-2.

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

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

$$x-2=4-4x+x^{2}$$

Subtract 4 from both sides.

$$x-2-4=-4x+x^{2}$$

Subtract 4 from -2 to get -6.

$$x-6=-4x+x^{2}$$

Add 4x to both sides.

$$x-6+4x=x^{2}$$

Combine x and 4x to get 5x.

$$5x-6=x^{2}$$

Subtract x^{2} from both sides.

$$5x-6-x^{2}=0$$

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

$$-x^{2}+5x-6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=5 ab=-\left(-6\right)=6$$

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 6.

$$1,6 2,3$$

Calculate the sum for each pair.

$$1+6=7 2+3=5$$

The solution is the pair that gives sum 5.

$$a=3 b=2$$

Rewrite -x^{2}+5x-6 as \left(-x^{2}+3x\right)+\left(2x-6\right).

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

Factor out -x in the first and 2 in the second group.

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

Factor out common term x-3 by using distributive property.

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

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

$$x=3 x=2$$

Substitute 3 for x in the equation 3-\sqrt{x-2}=5-x.

$$3-\sqrt{3-2}=5-3$$

Simplify. The value x=3 satisfies the equation.

$$2=2$$

Substitute 2 for x in the equation 3-\sqrt{x-2}=5-x.

$$3-\sqrt{2-2}=5-2$$

Simplify. The value x=2 satisfies the equation.

$$3=3$$

List all solutions of -\sqrt{x-2}=2-x.

$$x=3 x=2$$