Solve x-sqrt{4-x}=2

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

Solve for x

$$x=3$$

Solution Steps

Subtract x from both sides of the equation.

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

Square both sides of the equation.

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

Expand \left(-\sqrt{4-x}\right)^{2}.

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

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

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

Calculate \sqrt{4-x} to the power of 2 and get 4-x.

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

Use the distributive property to multiply 1 by 4-x.

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

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

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

Subtract 4 from both sides.

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

Subtract 4 from 4 to get 0.

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

Add 4x to both sides.

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

Combine -x and 4x to get 3x.

$$3x=x^{2}$$

Subtract x^{2} from both sides.

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

Factor out x.

$$x\left(3-x\right)=0$$

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

$$x=0 x=3$$

Substitute 0 for x in the equation x-\sqrt{4-x}=2.

$$0-\sqrt{4-0}=2$$

Simplify. The value x=0 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2=2$$

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

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

Simplify. The value x=3 satisfies the equation.

$$2=2$$

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

$$x=3$$