Solve sqrt{5-x}+1=x-2

$$\sqrt { 5 - x } + 1 = x - 2$$

Solve for x

$$x=4$$

Solution Steps

Subtract 1 from both sides of the equation.

$$\sqrt{5-x}=x-2-1$$

Subtract 1 from -2 to get -3.

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

Square both sides of the equation.

$$\left(\sqrt{5-x}\right)^{2}=\left(x-3\right)^{2}$$

Calculate \sqrt{5-x} to the power of 2 and get 5-x.

$$5-x=\left(x-3\right)^{2}$$

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

$$5-x=x^{2}-6x+9$$

Subtract x^{2} from both sides.

$$5-x-x^{2}=-6x+9$$

Add 6x to both sides.

$$5-x-x^{2}+6x=9$$

Combine -x and 6x to get 5x.

$$5+5x-x^{2}=9$$

Subtract 9 from both sides.

$$5+5x-x^{2}-9=0$$

Subtract 9 from 5 to get -4.

$$-4+5x-x^{2}=0$$

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

$$-x^{2}+5x-4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=5 ab=-\left(-4\right)=4$$

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 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 5.

$$a=4 b=1$$

Rewrite -x^{2}+5x-4 as \left(-x^{2}+4x\right)+\left(x-4\right).

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

Factor out -x in -x^{2}+4x.

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

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

$$\left(x-4\right)\left(-x+1\right)$$

To find equation solutions, solve x-4=0 and -x+1=0.

$$x=4 x=1$$

Substitute 4 for x in the equation \sqrt{5-x}+1=x-2.

$$\sqrt{5-4}+1=4-2$$

Simplify. The value x=4 satisfies the equation.

$$2=2$$

Substitute 1 for x in the equation \sqrt{5-x}+1=x-2.

$$\sqrt{5-1}+1=1-2$$

Simplify. The value x=1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$3=-1$$

Equation \sqrt{5-x}=x-3 has a unique solution.

$$x=4$$