Solve x-sqrt{x-2}=4

$$x - \sqrt { x - 2 } = 4$$

Solve for x

$$x=6$$

Solution Steps

Subtract x from both sides of the equation.

$$-\sqrt{x-2}=4-x$$

Square both sides of the equation.

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

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

$$\left(-1\right)^{2}\left(\sqrt{x-2}\right)^{2}=\left(4-x\right)^{2}$$

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

$$1\left(\sqrt{x-2}\right)^{2}=\left(4-x\right)^{2}$$

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

$$1\left(x-2\right)=\left(4-x\right)^{2}$$

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-x\right)^{2}.

$$x-2=16-8x+x^{2}$$

Subtract 16 from both sides.

$$x-2-16=-8x+x^{2}$$

Subtract 16 from -2 to get -18.

$$x-18=-8x+x^{2}$$

Add 8x to both sides.

$$x-18+8x=x^{2}$$

Combine x and 8x to get 9x.

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

Subtract x^{2} from both sides.

$$9x-18-x^{2}=0$$

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

$$-x^{2}+9x-18=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-18. To find a and b, set up a system to be solved.

$$a+b=9 ab=-\left(-18\right)=18$$

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 18.

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

Calculate the sum for each pair.

$$1+18=19 2+9=11 3+6=9$$

The solution is the pair that gives sum 9.

$$a=6 b=3$$

Rewrite -x^{2}+9x-18 as \left(-x^{2}+6x\right)+\left(3x-18\right).

$$\left(-x^{2}+6x\right)+\left(3x-18\right)$$

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

$$-x\left(x-6\right)+3\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(-x+3\right)$$

To find equation solutions, solve x-6=0 and -x+3=0.

$$x=6 x=3$$

Substitute 6 for x in the equation x-\sqrt{x-2}=4.

$$6-\sqrt{6-2}=4$$

Simplify. The value x=6 satisfies the equation.

$$4=4$$

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

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

Simplify. The value x=3 does not satisfy the equation.

$$2=4$$

Equation -\sqrt{x-2}=4-x has a unique solution.

$$x=6$$