Solve x-2=-sqrt{x-3}+1

$$x - 2 = - \sqrt { x - 3 } + 1$$

Solve for x

$$x=3$$

Solution Steps

Add \sqrt{x-3} to both sides.

$$x-2+\sqrt{x-3}=1$$

Add 2 to both sides.

$$x+\sqrt{x-3}=1+2$$

Add 1 and 2 to get 3.

$$x+\sqrt{x-3}=3$$

Subtract x from both sides of the equation.

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

Square both sides of the equation.

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

Calculate \sqrt{x-3} to the power of 2 and get x-3.

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

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

$$x-3=9-6x+x^{2}$$

Add 6x to both sides.

$$x-3+6x=9+x^{2}$$

Combine x and 6x to get 7x.

$$7x-3=9+x^{2}$$

Subtract x^{2} from both sides.

$$7x-3-x^{2}=9$$

Subtract 9 from both sides.

$$7x-3-x^{2}-9=0$$

Subtract 9 from -3 to get -12.

$$7x-12-x^{2}=0$$

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

$$-x^{2}+7x-12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=7 ab=-\left(-12\right)=12$$

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 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 7.

$$a=4 b=3$$

Rewrite -x^{2}+7x-12 as \left(-x^{2}+4x\right)+\left(3x-12\right).

$$\left(-x^{2}+4x\right)+\left(3x-12\right)$$

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

$$-x\left(x-4\right)+3\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(-x+3\right)$$

To find equation solutions, solve x-4=0 and -x+3=0.

$$x=4 x=3$$

Substitute 4 for x in the equation x-2=-\sqrt{x-3}+1.

$$4-2=-\sqrt{4-3}+1$$

Simplify. The value x=4 does not satisfy the equation.

$$2=0$$

Substitute 3 for x in the equation x-2=-\sqrt{x-3}+1.

$$3-2=-\sqrt{3-3}+1$$

Simplify. The value x=3 satisfies the equation.

$$1=1$$

Equation \sqrt{x-3}=3-x has a unique solution.

$$x=3$$