Solve sqrt{2x-4}-sqrt{x}=4

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

Solve for x

$$x=100$$

Solution Steps

Subtract -\sqrt{x} from both sides of the equation.

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

Square both sides of the equation.

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

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

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

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

$$2x-4=16+8\sqrt{x}+\left(\sqrt{x}\right)^{2}$$

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

$$2x-4=16+8\sqrt{x}+x$$

Subtract 16+x from both sides of the equation.

$$2x-4-\left(16+x\right)=8\sqrt{x}$$

To find the opposite of 16+x, find the opposite of each term.

$$2x-4-16-x=8\sqrt{x}$$

Subtract 16 from -4 to get -20.

$$2x-20-x=8\sqrt{x}$$

Combine 2x and -x to get x.

$$x-20=8\sqrt{x}$$

Square both sides of the equation.

$$\left(x-20\right)^{2}=\left(8\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-20\right)^{2}.

$$x^{2}-40x+400=\left(8\sqrt{x}\right)^{2}$$

Expand \left(8\sqrt{x}\right)^{2}.

$$x^{2}-40x+400=8^{2}\left(\sqrt{x}\right)^{2}$$

Calculate 8 to the power of 2 and get 64.

$$x^{2}-40x+400=64\left(\sqrt{x}\right)^{2}$$

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

$$x^{2}-40x+400=64x$$

Subtract 64x from both sides.

$$x^{2}-40x+400-64x=0$$

Combine -40x and -64x to get -104x.

$$x^{2}-104x+400=0$$

To solve the equation, factor x^{2}-104x+400 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-104 ab=400$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 400.

$$-1,-400 -2,-200 -4,-100 -5,-80 -8,-50 -10,-40 -16,-25 -20,-20$$

Calculate the sum for each pair.

$$-1-400=-401 -2-200=-202 -4-100=-104 -5-80=-85 -8-50=-58 -10-40=-50 -16-25=-41 -20-20=-40$$

The solution is the pair that gives sum -104.

$$a=-100 b=-4$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-100\right)\left(x-4\right)$$

To find equation solutions, solve x-100=0 and x-4=0.

$$x=100 x=4$$

Substitute 100 for x in the equation \sqrt{2x-4}-\sqrt{x}=4.

$$\sqrt{2\times 100-4}-\sqrt{100}=4$$

Simplify. The value x=100 satisfies the equation.

$$4=4$$

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

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

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

$$0=4$$

Substitute 100 for x in the equation \sqrt{2x-4}-\sqrt{x}=4.

$$\sqrt{2\times 100-4}-\sqrt{100}=4$$

Simplify. The value x=100 satisfies the equation.

$$4=4$$

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

$$x=100$$